Essentials of Robust Control

ESSENTIALS OF ROBUST CONTROL Kemin Zhou May 25, 1999 Preface Robustness of control systems to disturbances and uncertainties has always been the central issue in feedback control. Feedback would not be needed for most control systems if there were no disturbances and uncertainties. Developing multivariable robust control methods has been the focal point in the last two decades in the control community. The state-of-the-art H∞ robust control theory is the result of this effort. This book introduces some essentials of robust and H∞ control theory. It grew from another book by this author, John C. Doyle, and Keith Glover, entitled Robust and Optimal Control, which has been extensively class-tested in many universities around the world. Unlike that book, which is intended primarily as a comprehensive reference of robust and H∞ control theory, this book is intended to be a text for a graduate course in multivariable control. It is also intended to be a reference for practicing control engineers who are interested in applying the state-of-the-art robust control techniques in their applications. With this objective in mind, I have streamlined the presentation, added more than 50 illustrative examples, included many related Matlab R commands1 and more than 150 exercise problems, and added some recent developments in the area of robust control such as gap metric, ν-gap metric, model validation, and mixed µ problem. In addition, many proofs are completely rewritten and some advanced topics are either deleted completely or do not get an in-depth treatment. The prerequisite for reading this book is some basic knowledge of classical control theory and state-space theory. The text contains more material than could be covered in detail in a one-semester or a one-quarter course. Chapter 1 gives a chapter-by-chapter summary of the main results presented in the book, which could be used as a guide for the selection of topics for a specific course. Chapters 2 and 3 can be used as a refresher for some linear algebra facts and some standard linear system theory. A course focusing on H∞ control should cover at least most parts of Chapters 4–6, 8, 9, 11–13, and Sections 14.1 and 14.2. An advanced H∞ control course should also include the rest of Chapter 14, Chapter 16, and possibly Chapters 10, 7, and 15. A course focusing on robustness and model uncertainty should cover at least Chapters 4, 5, and 8–10. Chapters 17 and 18 can be added to any advanced robust and H∞ control course if time permits. I have tried hard to eliminate obvious mistakes. It is, however, impossible for me to make the book perfect. Readers are encouraged to send corrections, comments, and 1 Matlab is a registered trademark of The MathWorks, Inc. vii viii PREFACE suggestions to me, preferably by electronic mail, at kemin@ee.lsu.edu I am also planning to put any corrections, modifications, and extensions on the Internet so that they can be obtained either from the following anonymous ftp: ftp gate.ee.lsu.edu cd pub/kemin/books/essentials/ or from the author’s home page: http://kilo.ee.lsu.edu/kemin/books/essentials/ This book would not be possible without the work done jointly for the previous book with Professor John C. Doyle and Professor Keith Glover. I thank them for their influence on my research and on this book. Their serious attitudes toward scientific research have been reference models for me. I am especially grateful to John for having me as a research fellow in Caltech, where I had two very enjoyable years and had opportunities to catch a glimpse of his “BIG PICTURE” of control. I want to thank my editor from Prentice Hall, Tom Robbins, who originally proposed the idea for this book and has been a constant source of support for me while writing it. Without his support and encouragement, this project would have been a difficult one. It has been my great pleasure to work with him. I would like to express my sincere gratitude to Professor Bruce A. Francis for giving me many helpful comments and suggestions on this book. Professor Francis has also kindly provided many exercises in the book. I am also grateful to Professor Kang-Zhi Liu and Professor Zheng-Hua Luo, who have made many useful comments and suggestions. I want to thank Professor Glen Vinnicombe for his generous help in the preparation of Chapters 16 and 17. Special thanks go to Professor Jianqing Mao for providing me the opportunity to present much of this material in a series of lectures at Beijing University of Aeronautics and Astronautics in the summer of 1996. In addition, I would like to thank all those who have helped in many ways in making this book possible, especially Professor Pramod P. Khargonekar, Professor André Tits, Professor Andrew Packard, Professor Jie Chen, Professor Jakob Stoustrup, Professor Hans Henrik Niemann, Professor Malcolm Smith, Professor Tryphon Georgiou, Professor Tongwen Chen, Professor Hitay Özbay, Professor Gary Balas, Professor Carolyn Beck, Professor Dennis S. Bernstein, Professor Mohamed Darouach, Dr. Bobby Bodenheimer, Professor Guoxiang Gu, Dr. Weimin Lu, Dr. John Morris, Dr. Matt Newlin, Professor Li Qiu, Professor Hector P. Rotstein, Professor Andrew Teel, Professor Jagannathan Ramanujam, Dr. Linda G. Bushnell, Xiang Chen, Greg Salomon, Pablo A. Parrilo, and many other people. I would also like to thank the following agencies for supporting my research: National Science Foundation, Army Research Office (ARO), Air Force of Scientific Research, and the Board of Regents in the State of Louisiana. Finally, I would like to thank my wife, Jing, and my son, Eric, for their generous support, understanding, and patience during the writing of this book. Kemin Zhou PREFACE ix Here is how H∞ is pronounced in Chinese: It means “The joy of love is endless.” Contents Preface vii Notation and Symbols xv List of Acronyms 1 Introduction 1.1 What Is This Book About? 1.2 Highlights of This Book . . 1.3 Notes and References . . . . 1.4 Problems . . . . . . . . . . xvii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 9 10 2 Linear Algebra 2.1 Linear Subspaces . . . . . . . . . 2.2 Eigenvalues and Eigenvectors . . 2.3 Matrix Inversion Formulas . . . . 2.4 Invariant Subspaces . . . . . . . 2.5 Vector Norms and Matrix Norms 2.6 Singular Value Decomposition . . 2.7 Semidefinite Matrices . . . . . . 2.8 Notes and References . . . . . . . 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 12 13 15 16 19 23 24 24 . . . . . . . . 27 27 28 31 34 35 38 41 42 . . . . . . . . 3 Linear Systems 3.1 Descriptions of Linear Dynamical Systems . . . 3.2 Controllability and Observability . . . . . . . . 3.3 Observers and Observer-Based Controllers . . . 3.4 Operations on Systems . . . . . . . . . . . . . . 3.5 State-Space Realizations for Transfer Matrices 3.6 Multivariable System Poles and Zeros . . . . . 3.7 Notes and References . . . . . . . . . . . . . . . 3.8 Problems . . . . . . . . . . . . . . . . . . . . . xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 4 H2 4.1 4.2 4.3 4.4 4.5 4.6 CONTENTS and H∞ Spaces Hilbert Spaces . . . . . . . . . H2 and H∞ Spaces . . . . . . . Computing L2 and H2 Norms . Computing L∞ and H∞ Norms Notes and References . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 47 53 55 61 62 5 Internal Stability 5.1 Feedback Structure . . . . . . . . 5.2 Well-Posedness of Feedback Loop 5.3 Internal Stability . . . . . . . . . 5.4 Coprime Factorization over RH∞ 5.5 Notes and References . . . . . . . 5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 65 66 68 71 77 77 6 Performance Specifications and Limitations 6.1 Feedback Properties . . . . . . . . . . . . . . 6.2 Weighted H2 and H∞ Performance . . . . . . 6.3 Selection of Weighting Functions . . . . . . . 6.4 Bode’s Gain and Phase Relation . . . . . . . 6.5 Bode’s Sensitivity Integral . . . . . . . . . . . 6.6 Analyticity Constraints . . . . . . . . . . . . 6.7 Notes and References . . . . . . . . . . . . . . 6.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 . 81 . 85 . 89 . 94 . 98 . 100 . 102 . 102 7 Balanced Model Reduction 7.1 Lyapunov Equations . . . . . . . . . . . . . . . . 7.2 Balanced Realizations . . . . . . . . . . . . . . . 7.3 Model Reduction by Balanced Truncation . . . . 7.4 Frequency-Weighted Balanced Model Reduction . 7.5 Notes and References . . . . . . . . . . . . . . . . 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 106 107 117 124 126 127 8 Uncertainty and Robustness 8.1 Model Uncertainty . . . . . . . . . . . . . . 8.2 Small Gain Theorem . . . . . . . . . . . . . 8.3 Stability under Unstructured Uncertainties 8.4 Robust Performance . . . . . . . . . . . . . 8.5 Skewed Specifications . . . . . . . . . . . . 8.6 Classical Control for MIMO Systems . . . . 8.7 Notes and References . . . . . . . . . . . . . 8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 129 137 141 147 150 154 157 158 . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS xiii 9 Linear Fractional Transformation 9.1 Linear Fractional Transformations 9.2 Basic Principle . . . . . . . . . . . 9.3 Redheffer Star Products . . . . . . 9.4 Notes and References . . . . . . . . 9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 165 173 178 180 181 10 µ and µ Synthesis 10.1 General Framework for System Robustness . 10.2 Structured Singular Value . . . . . . . . . . . 10.3 Structured Robust Stability and Performance 10.4 Overview of µ Synthesis . . . . . . . . . . . . 10.5 Notes and References . . . . . . . . . . . . . . 10.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 184 187 200 213 216 217 11 Controller Parameterization 11.1 Existence of Stabilizing Controllers . . . . . . . 11.2 Parameterization of All Stabilizing Controllers 11.3 Coprime Factorization Approach . . . . . . . . 11.4 Notes and References . . . . . . . . . . . . . . . 11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 222 224 228 231 231 12 Algebraic Riccati Equations 12.1 Stabilizing Solution and Riccati 12.2 Inner Functions . . . . . . . . . 12.3 Notes and References . . . . . . 12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 234 245 246 246 13 H2 Optimal Control 13.1 Introduction to Regulator Problem . . 13.2 Standard LQR Problem . . . . . . . . 13.3 Extended LQR Problem . . . . . . . . 13.4 Guaranteed Stability Margins of LQR 13.5 Standard H2 Problem . . . . . . . . . 13.6 Stability Margins of H2 Controllers . . 13.7 Notes and References . . . . . . . . . . 13.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 253 255 258 259 261 265 267 267 14 H∞ Control 14.1 Problem Formulation . . . . . . . . 14.2 A Simplified H∞ Control Problem 14.3 Optimality and Limiting Behavior 14.4 Minimum Entropy Controller . . . 14.5 An Optimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 269 270 282 286 286 . . . . . . . . . . xiv CONTENTS 14.6 General H∞ Solutions . . . . 14.7 Relaxing Assumptions . . . . 14.8 H2 and H∞ Integral Control 14.9 H∞ Filtering . . . . . . . . . 14.10Notes and References . . . . . 14.11Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 291 294 297 299 300 15 Controller Reduction 305 15.1 H∞ Controller Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . 306 15.2 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 15.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 16 H∞ Loop Shaping 16.1 Robust Stabilization of Coprime Factors 16.2 Loop-Shaping Design . . . . . . . . . . . 16.3 Justification for H∞ Loop Shaping . . . 16.4 Further Guidelines for Loop Shaping . . 16.5 Notes and References . . . . . . . . . . . 16.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 315 325 328 334 341 342 17 Gap Metric and ν-Gap Metric 17.1 Gap Metric . . . . . . . . . . . . . . . . . 17.2 ν-Gap Metric . . . . . . . . . . . . . . . . 17.3 Geometric Interpretation of ν-Gap Metric 17.4 Extended Loop-Shaping Design . . . . . . 17.5 Controller Order Reduction . . . . . . . . 17.6 Notes and References . . . . . . . . . . . . 17.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 350 357 370 373 375 375 375 18 Miscellaneous Topics 18.1 Model Validation . . . . . . . . 18.2 Mixed µ Analysis and Synthesis 18.3 Notes and References . . . . . . 18.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 377 381 389 390 . . . . . . . . . . . . . . . . . . . . . . . . Bibliography 391 Index 407 Notation and Symbols R and C F C− and C− C+ and C+ jR fields of real and complex numbers field, either R or C open and closed left-half plane open and closed right-half plane imaginary axis ∈ ⊂ ∪ ∩ belong to subset union intersection ✷ ✸ end of proof end of remark := defined as ' and / asymptotically greater and less than ≫ and ≪ much greater and less than α |α| Re(α) complex conjugate of α ∈ C absolute value of α ∈ C real part of α ∈ C In [aij ] diag(a1 , . . . , an ) AT and A∗ A−1 and A+ A−∗ det(A) trace(A) n × n identity matrix a matrix with aij as its ith row and jth column element an n × n diagonal matrix with ai as its ith diagonal element transpose and complex conjugate transpose of A inverse and pseudoinverse of A shorthand for (A−1 )∗ determinant of A trace of A xv xvi NOTATION AND SYMBOLS λ(A) ρ(A) ρR (A) σ(A) and σ(A) σi (A) κ(A) kAk Im(A), R(A) Ker(A), N(A) X− (A) eigenvalue of A spectral radius of A real spectrum radius of A the largest and the smallest singular values of A ith singular value of A condition number of A spectral norm of A: kAk = σ(A) image (or range) space of A kernel (or null) space of A stable invariant subspace of A Ric(H) g∗f ∠ h, i x⊥y D⊥ S⊥ the stabilizing solution of an ARE convolution of g and f angle inner product orthogonal, hx, yi = 0 orthogonal complement of D orthogonal complement of subspace S, e.g., H2⊥ L2 (−∞, ∞) L2+ := L2 [0, ∞) L2− := L2 (−∞, 0] L2 (jR) H2 H2⊥ L∞ (jR) H∞ − H∞ time domain square integrable functions subspace of L2 (−∞, ∞) with functions zero for t < 0 subspace of L2 (−∞, ∞) with functions zero for t > 0 square integrable functions on C0 including at ∞ subspace of L2 (jR) with functions analytic in Re(s) > 0 subspace of L2 (jR) with functions analytic in Re(s) < 0 functions bounded on Re(s) = 0 including at ∞ the set of L∞ (jR) functions analytic in Re(s) > 0 the set of L∞ (jR) functions analytic in Re(s) < 0 prefix B and Bo prefix R closed and open unit ball, e.g. B∆ and Bo ∆ real rational, e.g., RH∞ and RH2 , etc. Rp (s) rational proper transfer matrices ∼ G  (s)  A B C D η(G(s)) η0 (G(s)) wno(G) shorthand for GT (−s) Fℓ (M, Q) Fu (M, Q) M ⋆N shorthand for state space realization C(sI − A)−1 B + D number of right-half plane poles number of imaginary axis poles winding number lower LFT upper LFT star product List of Acronyms ARE FDLTI iff lcf LFT lhp or LHP LQG LTI MIMO nlcf NP nrcf NS rcf rhp or RHP RP RS SISO SSV SVD algebraic Riccati equation finite dimensional linear time invariant if and only if left coprime factorization linear fractional transformation left-half plane Re(s) < 0 linear quadratic Gaussian linear time invariant multi-input multioutput normalized left coprime factorization nominal performance normalized right coprime factorization nominal stability right coprime factorization right-half plane Re(s) > 0 robust performance robust stability single-input single-output structured singular value (µ) singular value decomposition xvii xviii LIST OF ACRONYMS Chapter 1 Introduction This chapter gives a brief description of the problems considered in this book and the key results presented in each chapter. 1.1 What Is This Book About? This book is about basic robust and H∞ control theory. We consider a control system with possibly multiple sources of uncertainties, noises, and disturbances as shown in Figure 1.1. uncertainty disturbance other controlled signals uncertainty System Interconnection tracking errors uncertainty noise controller reference signals Figure 1.1: General system interconnection 1 2 INTRODUCTION We consider mainly two types of problems: • Analysis problems: Given a controller, determine if the controlled signals (including tracking errors, control signals, etc.) satisfy the desired properties for all admissible noises, disturbances, and model uncertainties. • Synthesis problems: Design a controller so that the controlled signals satisfy the desired properties for all admissible noises, disturbances, and model uncertainties. Most of our analysis and synthesis will be done on a unified linear fractional transformation (LFT) framework. To that end, we shall show that the system shown in Figure 1.1 can be put in the general diagram in Figure 1.2, where P is the interconnection matrix, K is the controller, ∆ is the set of all possible uncertainty, w is a vector signal including noises, disturbances, and reference signals, z is a vector signal including all controlled signals and tracking errors, u is the control signal, and y is the measurement. ✲ ∆ v z✛ y P ✲ K ✛ ✛ ✛ η w u Figure 1.2: General LFT framework The block diagram in Figure 1.2 represents the following equations:     v η  z  = P w  y u η = ∆v u = Ky. Let the transfer matrix from w to z be denoted by Tzw and assume that the admissible uncertainty ∆ satisfies k∆k∞ < 1/γu for some γu > 0. Then our analysis problem is to answer if the closed-loop system is stable for all admissible ∆ and kTzw k∞ ≤ γp for some prespecified γp > 0, where kTzw k∞ is the H∞ norm defined as kTzw k∞ = supω σ̄ (Tzw (jω)). The synthesis problem is to design a controller K so that the aforementioned robust stability and performance conditions are satisfied. In the simplest form, we have either ∆ = 0 or w = 0. The former becomes the wellknown H∞ control problem and the later becomes the robust stability problem. The two 1.2. Highlights of This Book 3 problems are equivalent when ∆ is a single-block unstructured uncertainty through the application of the small gain theorem (see Chapter 8). This robust stability consequence was probably the main motivation for the development of H∞ methods. The analysis and synthesis for systems with multiple-block ∆ can be reduced in most cases to an equivalent H∞ problem with suitable scalings. Thus a solution to the H∞ control problem is the key to all robustness problems considered in this book. In the next section, we shall give a chapter-by-chapter summary of the main results presented in this book. We refer readers to the book Robust and Optimal Control by K. Zhou, J. C. Doyle, and K. Glover [1996] for a brief historical review of H∞ and robust control and for some detailed treatment of some advanced topics. 1.2 Highlights of This Book The key results in each chapter are highlighted in this section. Readers should consult the corresponding chapters for the exact statements and conditions. Chapter 2 reviews some basic linear algebra facts. Chapter 3 reviews system theoretical concepts: controllability, observability, stabilizability, detectability, pole placement, observer theory, system poles and zeros, and state-space realizations. Chapter 4 introduces the H2 spaces and the H∞ spaces. State-space methods of computing real rational H2 and H∞ transfer matrix norms are presented. For example, let   A B ∈ RH∞ . G(s) = C 0 Then kGk22 = trace(B ∗ QB) = trace(CP C ∗ ) and kGk∞ = max{γ : H has an eigenvalue on the imaginary axis}, where P and Q are the controllability and observability Gramians and   A BB ∗ /γ 2 H= . −C ∗ C −A∗ 4 INTRODUCTION Chapter 5 introduces the feedback structure and discusses its stability. w1 e1 ✲e + ✻ + ✲ P K̂ ✛ e2 + w2 ❄ ✛+ e We define that the above closed-loop system is internally stable if and only if  I −P −K̂ I −1 =  (I − K̂P )−1 P (I − K̂P )−1 K̂(I − P K̂)−1 (I − P K̂)−1  ∈ RH∞ . Alternative characterizations of internal stability using coprime factorizations are also presented. Chapter 6 considers the feedback system properties and design limitations. The formulations of optimal H2 and H∞ control problems and the selection of weighting functions are also considered in this chapter. Chapter 7 considers the problem of reducing the order of a linear multivariable dynamical system using the balanced truncation method. Suppose   B1 A11 A12 B2  ∈ RH∞ G(s) =  A21 A22 C1 C2 D is a balanced realization with controllability and observability Gramians P = Q = Σ = diag(Σ1 , Σ2 ) = diag(σ1 Is1 , σ2 Is2 , . . . , σr Isr ) = diag(σr+1 Isr+1 , σr+2 Isr+2 , . . . , σN IsN ).   A11 B1 Then the truncated system Gr (s) = is stable and satisfies an additive C1 D error bound: N X σi . kG(s) − Gr (s)k∞ ≤ 2 Σ1 Σ2 i=r+1 Frequency-weighted balanced truncation method is also discussed. Chapter 8 derives robust stability tests for systems under various modeling assumptions through the use of the small gain theorem. In particular, we show that a system, shown at the top of the following page, with an unstructured uncertainty ∆ ∈ RH∞ 1.2. Highlights of This Book 5 with k∆k∞ < 1 is robustly stable if and only if kTzw k∞ ≤ 1, where Tzw is the matrix transfer function from w to z. ∆ z w nominal system Chapter 9 introduces the LFT in detail. We show that many control problems can be formulated and treated in the LFT framework. In particular, we show that every analysis problem can be put in an LFT form with some structured ∆(s) and some interconnection matrix M (s) and every synthesis problem can be put in an LFT form with a generalized plant G(s) and a controller K(s) to be designed. z ✛ ✲ ∆ z ✛ M ✛ ✛ G y w ✲K w ✛ ✛ u Chapter 10 considers robust stability and performance for systems with multiple sources of uncertainties. We show that an uncertain system is robustly stable and satisfies some H∞ performance criterion for all ∆i ∈ RH∞ with k∆i k∞ < 1 if and only if the structured singular value (µ) of the corresponding interconnection model is no greater than 1. ∆1 ∆4 nominal system ∆3 ∆2 6 INTRODUCTION Chapter 11 characterizes in state-space all controllers that stabilize a given dynamical system G(s). For a given generalized plant     B2 A B1 G11 (s) G12 (s) G(s) = =  C1 D11 D12  G21 (s) G22 (s) C2 D21 D22 we show that all stabilizing controllers can be parameterized as the transfer matrix from y to u below where F and L are such that A + LC2 and A + B2 F are stable and where Q is any stable proper transfer matrix. ✛ z G w ✛ ✛ y u D22 ✛ ❄ c✛ ❄ c✛ − C2 ✛ R ✛ c✛ c✛ B2 ✛ ✻✻ c✛ ✻ ✲ A ✲ F ✲ −L y1 u1 ✲ Q Chapter 12 studies the stabilizing solution to an algebraic Riccati equation (ARE). A solution to the following ARE A∗ X + XA + XRX + Q = 0 is said to be a stabilizing solution if A + RX is stable. Now let   A R H := −Q −A∗ and let X− (H) be the stable H invariant subspace and   X1 X− (H) = Im , X2 where X1 , X2 ∈ Cn×n . If X1 is nonsingular, then X := X2 X1−1 is uniquely determined by H, denoted by X = Ric(H). A key result of this chapter is the so-called bounded 1.2. Highlights of This Book 7 real lemma, which states that a stable transfer matrix G(s) satisfies kG(s)k∞ < γ if and only if there exists an X such that A + BB ∗ X/γ 2 is stable and XA + A∗ X + XBB ∗ X/γ 2 + C ∗ C = 0. The H∞ control theory in Chapter 14 will be derived based on this lemma. Chapter 13 treats the optimal control of linear time-invariant systems with quadratic performance criteria (i.e., H2 problems). We consider a dynamical system described by an LFT with   B2 A B1 G(s) =  C1 0 D12  . 0 C2 D21 ✛z ✛w G ✛ y u ✲ K Define ∗ R1 = D12 D12 > 0, H2 := J2 := F2 :=   ∗ R2 = D21 D21 >0 ∗ A − B2 R1−1 D12 C1 −1 ∗ ∗ −C1 (I − D12 R1 D12 )C1 ∗ (A − B1 D21 R2−1 C2 )∗ ∗ −B1 (I − D21 R2−1 D21 )B1∗ X2 := Ric(H2 ) ≥ 0, −R1−1 (B2∗ X2 + ∗ D12 C1 ), −B2 R1−1 B2∗ ∗ −(A − B2 R1−1 D12 C1 )∗ −C2∗ R2−1 C2 ∗ −(A − B1 D21 R2−1 C2 )   Y2 := Ric(J2 ) ≥ 0 ∗ L2 := −(Y2 C2∗ + B1 D21 )R2−1 . Then the H2 optimal controller (i.e., the controller that minimizes kTzw k2 ) is given by   A + B2 F2 + L2 C2 −L2 . Kopt (s) := F2 0 Chapter 14 first considers an H∞ control problem with the generalized plant G(s) as given in Chapter 13 but with some additional simplifications: R1 = I, R2 = I, ∗ ∗ D12 C1 = 0, and B1 D21 = 0. We show that there exists an admissible controller such that kTzw k∞ < γ if and only if the following three conditions hold: (i) H∞ ∈ dom(Ric) and X∞ := Ric(H∞ ) > 0, where   A γ −2 B1 B1∗ − B2 B2∗ H∞ := ; −C1∗ C1 −A∗ 8 INTRODUCTION (ii) J∞ ∈ dom(Ric) and Y∞ := Ric(J∞ ) > 0, where   A∗ γ −2 C1∗ C1 − C2∗ C2 J∞ := ; −B1 B1∗ −A (iii) ρ(X∞ Y∞ ) < γ 2 . Moreover, an admissible controller such that kTzw k∞ < γ is given by   Â∞ −Z∞ L∞ Ksub = F∞ 0 where Â∞ := A + γ −2 B1 B1∗ X∞ + B2 F∞ + Z∞ L∞ C2 F∞ := −B2∗ X∞ , Z∞ := (I − γ −2 Y∞ X∞ )−1 . L∞ := −Y∞ C2∗ , We then consider further the general H∞ control problem. We indicate how various assumptions can be relaxed to accommodate other more complicated problems, such as singular control problems. We also consider the integral control in the H2 and H∞ theory and show how the general H∞ solution can be used to solve the H∞ filtering problem. Chapter 15 considers the design of reduced-order controllers by means of controller reduction. Special attention is paid to the controller reduction methods that preserve the closed-loop stability and performance. Methods are presented that give sufficient conditions in terms of frequency-weighted model reduction. Chapter 16 first solves a special H∞ minimization problem. Let P = M̃ −1 Ñ be a normalized left coprime factorization. Then we show that     K (I + P K)−1 I P inf I K stabilizing ∞ = inf K stabilizing  K I  −1 (I + P K) M̃ −1 = ∞ q 1− This implies that there is a robustly stabilizing controller for ˜ M )−1 (Ñ + ∆ ˜ N) P∆ = (M̃ + ∆ with if and only if  ǫ≤ ˜N ∆ ˜M ∆ q  1− Ñ  ∞ M̃ <ǫ  2 H .  Ñ M̃  2 H −1 . 1.3. Notes and References 9 Using this stabilization result, a loop-shaping design technique is proposed. The proposed technique uses only the basic concept of loop-shaping methods, and then a robust stabilization controller for the normalized coprime factor perturbed system is used to construct the final controller. Chapter 17 introduces the gap metric and the ν-gap metric. The frequency domain interpretation and applications of the ν-gap metric are discussed. The controller order reduction in the gap or ν-gap metric framework is also considered. Chapter 18 considers briefly the problems of model validation and the mixed real and complex µ analysis and synthesis. Most computations and examples in this book are done using Matlab. Since we shall use Matlab as a major computational tool, it is assumed that readers have some basic working knowledge of the Matlab operations (for example, how to input vectors and matrices). We have also included in this book some brief explanations of Matlab, Simulink R , Control System Toolbox, and µ Analysis and Synthesis Toolbox1 commands. In particular, this book is written consistently with the µ Analysis and Synthesis Toolbox. (Robust Control Toolbox, LMI Control Toolbox, and other software packages may equally be used with this book.) Thus it is helpful for readers to have access to this toolbox. It is suggested at this point to try the following demo programs from this toolbox. ≫ msdemo1 ≫ msdemo2 We shall introduce many more Matlab commands in the subsequent chapters. 1.3 Notes and References The original formulation of the H∞ control problem can be found in Zames [1981]. Relations between H∞ have now been established with many other topics in control: for example, risk-sensitive control of Whittle [1990]; differential games (see Başar and Bernhard [1991], Limebeer, Anderson, Khargonekar, and Green [1992]; Green and Limebeer [1995]); chain-scattering representation, and J-lossless factorization (Green [1992] and Kimura [1997]). See also Zhou, Doyle, and Glover [1996] for additional discussions and references. The state-space theory of H∞ has also been carried much further, by generalizing time invariant to time varying, infinite horizon to finite horizon, and finite dimensional to infinite dimensional, and even to some nonlinear settings. 1 Simulink is a registered trademark of The MathWorks, Inc.; µ-Analysis and Synthesis is a trademark of The MathWorks, Inc. and MUSYN Inc.; Control System Toolbox, Robust Control Toolbox, and LMI Control Toolbox are trademarks of The MathWorks, Inc. 10 1.4 INTRODUCTION Problems Problem 1.1 We shall solve an easy problem first. When you read a paper or a book, you often come across a statement like this “It is easy ...”. What the author really meant was one of the following: (a) it is really easy; (b) it seems to be easy; (c) it is easy for an expert; (d) the author does not know how to show it but he or she thinks it is correct. Now prove that when I say “It is easy” in this book, I mean it is really easy. (Hint: If you can prove it after you read the whole book, ask your boss for a promotion. If you cannot prove it after you read the whole book, trash the book and write a book yourself. Remember use something like “it is easy ...” if you are not sure what you are talking about.) Chapter 2 Linear Algebra Some basic linear algebra facts will be reviewed in this chapter. The detailed treatment of this topic can be found in the references listed at the end of the chapter. Hence we shall omit most proofs and provide proofs only for those results that either cannot be easily found in the standard linear algebra textbooks or are insightful to the understanding of some related problems. 2.1 Linear Subspaces Let R denote the real scalar field and C the complex scalar field. For the interest of this chapter, let F be either R or C and let Fn be the vector space over F (i.e., Fn is either Rn or Cn ). Now let x1 , x2 , . . . , xk ∈ Fn . Then an element of the form α1 x1 +. . .+αk xk with αi ∈ F is a linear combination over F of x1 , . . . , xk . The set of all linear combinations of x1 , x2 , . . . , xk ∈ Fn is a subspace called the span of x1 , x2 , . . . , xk , denoted by span{x1 , x2 , . . . , xk } := {x = α1 x1 + . . . + αk xk : αi ∈ F}. A set of vectors x1 , x2 , . . . , xk ∈ Fn is said to be linearly dependent over F if there exists α1 , . . . , αk ∈ F not all zero such that α1 x2 + . . . + αk xk = 0; otherwise the vectors are said to be linearly independent. Let S be a subspace of Fn , then a set of vectors {x1 , x2 , . . . , xk } ∈ S is called a basis for S if x1 , x2 , . . . , xk are linearly independent and S = span{x1 , x2 , . . . , xk }. However, such a basis for a subspace S is not unique but all bases for S have the same number of elements. This number is called the dimension of S, denoted by dim(S). A set of vectors {x1 , x2 , . . . , xk } in Fn is mutually orthogonal if x∗i xj = 0 for all i 6= j and orthonormal if x∗i xj = δij , where the superscript ∗ denotes complex conjugate transpose and δij is the Kronecker delta function with δij = 1 for i = j and δij = 0 for i 6= j. More generally, a collection of subspaces S1 , S2 , . . . , Sk of Fn is mutually orthogonal if x∗ y = 0 whenever x ∈ Si and y ∈ Sj for i 6= j. 11 12 LINEAR ALGEBRA The orthogonal complement of a subspace S ⊂ Fn is defined by S ⊥ := {y ∈ Fn : y ∗ x = 0 for all x ∈ S}. We call a set of vectors {u1 , u2 , . . . , uk } an orthonormal basis for a subspace S ∈ Fn if the vectors form a basis of S and are orthonormal. It is always possible to extend such a basis to a full orthonormal basis {u1 , u2 , . . . , un } for Fn . Note that in this case S ⊥ = span{uk+1 , . . . , un }, and {uk+1 , . . . , un } is called an orthonormal completion of {u1 , u2 , . . . , uk }. Let A ∈ Fm×n be a linear transformation from Fn to Fm ; that is, A : Fn 7−→ Fm . Then the kernel or null space of the linear transformation A is defined by KerA = N (A) := {x ∈ Fn : Ax = 0}, and the image or range of A is ImA = R(A) := {y ∈ Fm : y = Ax, x ∈ Fn }. Let ai , i = 1, 2, . . . , n denote the columns of a matrix A ∈ Fm×n ; then ImA = span{a1 , a2 , . . . , an }. A square matrix U ∈ F n×n whose columns form an orthonormal basis for Fn is called a unitary matrix (or orthogonal matrix if F = R), and it satisfies U ∗ U = I = U U ∗ . Now let A = [aij ] ∈ Cn×n ; then the trace of A is defined as trace(A) := n X aii . i=1 Illustrative MATLAB Commands: ≫ basis of KerA = null(A); basis of ImA = orth(A); rank of A = rank(A); 2.2 Eigenvalues and Eigenvectors Let A ∈ Cn×n ; then the eigenvalues of A are the n roots of its characteristic polynomial p(λ) = det(λI − A). The maximal modulus of the eigenvalues is called the spectral radius, denoted by ρ(A) := max |λi | 1≤i≤n if λi is a root of p(λ), where, as usual, | · | denotes the magnitude. The real spectral radius of a matrix A, denoted by ρR (A), is the maximum modulus of the real eigenvalues 2.3. Matrix Inversion Formulas 13 of A; that is, ρR (A) := max |λi | and ρR (A) := 0 if A has no real eigenvalues. A nonzero λi ∈R vector x ∈ Cn that satisfies Ax = λx is referred to as a right eigenvector of A. Dually, a nonzero vector y is called a left eigenvector of A if y ∗ A = λy ∗ . In general, eigenvalues need not be real, and neither do their corresponding eigenvectors. However, if A is real and λ is a real eigenvalue of A, then there is a real eigenvector corresponding to λ. In the case that all eigenvalues of a matrix A are real, we will denote λmax (A) for the largest eigenvalue of A and λmin (A) for the smallest eigenvalue. In particular, if A is a Hermitian matrix (i.e., A = A∗ ), then there exist a unitary matrix U and a real diagonal matrix Λ such that A = U ΛU ∗ , where the diagonal elements of Λ are the eigenvalues of A and the columns of U are the eigenvectors of A. Lemma 2.1 Consider the Sylvester equation AX + XB = C, (2.1) where A ∈ Fn×n , B ∈ Fm×m , and C ∈ Fn×m are given matrices. There exists a unique solution X ∈ Fn×m if and only if λi (A) + λj (B) 6= 0, ∀i = 1, 2, . . . , n, and j = 1, 2, . . . , m. In particular, if B = A∗ , equation (2.1) is called the Lyapunov equation; and the necessary and sufficient condition for the existence of a unique solution is that λi (A) + λ̄j (A) 6= 0, ∀i, j = 1, 2, . . . , n. Illustrative MATLAB Commands: ≫ [V, D] = eig(A) % AV = V D ≫ X=lyap(A,B,-C) 2.3 % solving Sylvester equation. Matrix Inversion Formulas Let A be a square matrix partitioned as follows:   A11 A12 , A := A21 A22 where A11 and A22 are also square matrices. Now suppose A11 is nonsingular; then A has the following decomposition:       I 0 A11 0 A11 A12 I A−1 A12 11 = −1 0 ∆ A21 A22 A21 A11 I 0 I 14 LINEAR ALGEBRA with ∆ := A22 − A21 A−1 11 A12 , and A is nonsingular iff ∆ is nonsingular. Dually, if A22 is nonsingular, then       −1 ˆ I 0 A11 A12 I A12 A22 ∆ 0 = A21 A22 A−1 I 0 I 0 A22 22 A21 ˆ := A11 − A12 A−1 A21 , and A is nonsingular iff ∆ ˆ is nonsingular. The matrix ∆ with ∆ 22 ˆ (∆) is called the Schur complement of A11 (A22 ) in A. Moreover, if A is nonsingular, then −1  −1  −1 −1 A11 + A11 A12 ∆−1 A21 A11 A11 A12 = −1 −1 A21 A22 −∆ A21 A11 and  A11 A21 A12 A22 −1 =  ˆ −1 ∆ −1 ˆ −1 −A22 A21 ∆ −1 −A−1 11 A12 ∆ −1 ∆ ˆ −1 A12 A−1 −∆ 22 −1 ˆ −1 A12 A−1 A22 + A−1 ∆ A 21 22 22   . The preceding matrix inversion formulas are particularly simple if A is block triangular: −1    −1 A11 0 A11 0 = −1 −1 A21 A22 −A22 A21 A11 A−1 22  −1  −1  −1 −1 A11 A12 A11 −A11 A12 A22 = . −1 0 A22 0 A22 The following identity is also very useful. Suppose A11 and A22 are both nonsingular matrices; then −1 −1 −1 −1 (A11 − A12 A−1 = A11 + A11 A12 (A22 − A21 A−1 A21 A−1 22 A21 ) 11 A12 ) 11 . As a consequence of the matrix decomposition formulas mentioned previously, we can calculate the determinant of a matrix by using its submatrices. Suppose A11 is nonsingular; then det A = det A11 det(A22 − A21 A−1 11 A12 ). On the other hand, if A22 is nonsingular, then −1 det A = det A22 det(A11 − A12 A22 A21 ). In particular, for any B ∈ Cm×n and C ∈ Cn×m , we have   Im B = det(In + CB) = det(Im + BC) det −C In and for x, y ∈ Cn det(In + xy ∗ ) = 1 + y ∗ x. Related MATLAB Commands: inv, det 2.4. Invariant Subspaces 2.4 15 Invariant Subspaces Let A : Cn 7−→ Cn be a linear transformation, λ be an eigenvalue of A, and x be a corresponding eigenvector, respectively. Then Ax = λx and A(αx) = λ(αx) for any α ∈ C. Clearly, the eigenvector x defines a one-dimensional subspace that is invariant with respect to premultiplication by A since Ak x = λk x, ∀k. In general, a subspace S ⊂ Cn is called invariant for the transformation A, or A-invariant, if Ax ∈ S for every x ∈ S. In other words, that S is invariant for A means that the image of S under A is contained in S: AS ⊂ S. For example, {0}, Cn , KerA, and ImA are all A-invariant subspaces. As a generalization of the one-dimensional invariant subspace induced by an eigenvector, let λ1 , . . . , λk be eigenvalues of A (not necessarily distinct), and let xi be the corresponding eigenvectors and the generalized eigenvectors. Then S = span{x1 , . . . , xk } is an A-invariant subspace provided that all the lower-rank generalized eigenvectors are included. More specifically, let λ1 = λ2 = · · · = λl be eigenvalues of A, and let x1 , x2 , . . . , xl be the corresponding eigenvector and the generalized eigenvectors obtained through the following equations: (A − λ1 I)x1 (A − λ1 I)x2 (A − λ1 I)xl = 0 = x1 .. . = xl−1 . Then a subspace S with xt ∈ S for some t ≤ l is an A-invariant subspace only if all lowerrank eigenvectors and generalized eigenvectors of xt are in S (i.e., xi ∈ S, ∀1 ≤ i ≤ t). This will be further illustrated in Example 2.1. On the other hand, if S is a nontrivial subspace1 and is A-invariant, then there is x ∈ S and λ such that Ax = λx. An A-invariant subspace S ⊂ Cn is called a stable invariant subspace if all the eigenvalues of A constrained to S have negative real parts. Stable invariant subspaces will play an important role in computing the stabilizing solutions to the algebraic Riccati equations in Chapter 12. Example 2.1 Suppose a matrix A has the following Jordan canonical form:   λ1 1      λ1  A x1 x2 x3 x4 = x1 x2 x3 x4    λ3 λ4 1 We will say subspace S is trivial if S = {0}. 16 LINEAR ALGEBRA with Reλ1 < 0, λ3 < 0, and λ4 > 0. Then it is easy to verify that S1 S3 S4 = span{x1 } = span{x3 } = span{x4 } S12 S13 S14 = span{x1 , x2 } = span{x1 , x3 } = span{x1 , x4 } S123 S124 S34 = span{x1 , x2 , x3 } = span{x1 , x2 , x4 } = span{x3 , x4 } are all A-invariant subspaces. Moreover, S1 , S3 , S12 , S13 , and S123 are stable A-invariant subspaces. The subspaces S2 = span{x2 }, S23 = span{x2 , x3 }, S24 = span{x2 , x4 }, and S234 = span{x2 , x3 , x4 } are, however, not A-invariant subspaces since the lower-rank eigenvector x1 is not in these subspaces. To illustrate, consider the subspace S23 . Then by definition, Ax2 ∈ S23 if it is an A-invariant subspace. Since Ax2 = λx2 + x1 , Ax2 ∈ S23 would require that x1 be a linear combination of x2 and x3 , but this is impossible since x1 is independent of x2 and x3 . 2.5 Vector Norms and Matrix Norms In this section, we shall define vector and matrix norms. Let X be a vector space. A real-valued function k·k defined on X is said to be a norm on X if it satisfies the following properties: (i) kxk ≥ 0 (positivity); (ii) kxk = 0 if and only if x = 0 (positive definiteness); (iii) kαxk = |α| kxk, for any scalar α (homogeneity); (iv) kx + yk ≤ kxk + kyk (triangle inequality) for any x ∈ X and y ∈ X. Let x ∈ Cn . Then we define the vector p-norm of x as !1/p n X |xi |p , for 1 ≤ p < ∞. kxkp := i=1 In particular, when p = 1, 2, ∞ we have kxk1 := n X i=1 |xi |; v u n uX kxk2 := t |xi |2 ; i=1 2.5. Vector Norms and Matrix Norms 17 kxk∞ := max |xi |. 1≤i≤n Clearly, norm is an abstraction and extension of our usual concept of length in threedimensional Euclidean space. So a norm of a vector is a measure of the vector “length” (for example, kxk2 is the Euclidean distance of the vector x from the origin). Similarly, we can introduce some kind of measure for a matrix. Let A = [aij ] ∈ Cm×n ; then the matrix norm induced by a vector p-norm is defined as kAxkp kAkp := sup . x6=0 kxkp The matrix norms induced by vector p-norms are sometimes called induced p-norms. This is because kAkp is defined by or induced from a vector p-norm. In fact, A can be viewed as a mapping from a vector space Cn equipped with a vector norm k·kp to another vector space Cm equipped with a vector norm k·kp . So from a system theoretical point of view, the induced norms have the interpretation of input/output amplification gains. In particular, the induced matrix 2-norm can be computed as p kAk2 = λmax (A∗ A). We shall adopt the following convention throughout this book for the vector and matrix norms unless specified otherwise: Let x ∈ Cn and A ∈ Cm×n ; then we shall denote the Euclidean 2-norm of x simply by kxk := kxk2 and the induced 2-norm of A by kAk := kAk2 . The Euclidean 2-norm has some very nice properties: Lemma 2.2 Let x ∈ Fn and y ∈ Fm . 1. Suppose n ≥ m. Then kxk = kyk iff there is a matrix U ∈ Fn×m such that x = U y and U ∗ U = I. 2. Suppose n = m. Then |x∗ y| ≤ kxk kyk. Moreover, the equality holds iff x = αy for some α ∈ F or y = 0. 3. kxk ≤ kyk iff there is a matrix ∆ ∈ Fn×m with k∆k ≤ 1 such that x = ∆y. Furthermore, kxk < kyk iff k∆k < 1. 4. kU xk = kxk for any appropriately dimensioned unitary matrices U . 18 LINEAR ALGEBRA Another often used matrix norm is the so called Frobenius norm. It is defined as v uX n p um X |aij |2 . kAkF := trace(A∗ A) = t i=1 j=1 However, the Frobenius norm is not an induced norm. The following properties of matrix norms are easy to show: Lemma 2.3 Let A and B be any matrices with appropriate dimensions. Then 1. ρ(A) ≤ kAk (this is also true for the F -norm and any induced matrix norm). 2. kABk ≤ kAk kBk. In particular, this gives A−1 (This is also true for any induced matrix norm.) ≥ kAk −1 if A is invertible. 3. kU AV k = kAk, and kU AV kF = kAkF , for any appropriately dimensioned unitary matrices U and V . 4. kABkF ≤ kAk kBkF and kABkF ≤ kBk kAkF . Note that although premultiplication or postmultiplication of a unitary matrix on a matrix does not change its induced 2-norm and F -norm, it does change its eigenvalues. For example, let   1 0 A= . 1 0 Then λ1 (A) = 1, λ2 (A) = 0. Now let U= " √1 2 − √12 √1 2 √1 2 # ; then U is a unitary matrix and   √ 2 0 UA = 0 0 √ with λ1 (U A) = 2, λ2 (U A) = 0. This property is useful in some matrix perturbation problems, particularly in the computation of bounds for structured singular values, which will be studied in Chapter 9. Related MATLAB Commands: norm, normest 2.6. Singular Value Decomposition 2.6 19 Singular Value Decomposition A very useful tool in matrix analysis is singular value decomposition (SVD). It will be seen that singular values of a matrix are good measures of the “size” of the matrix and that the corresponding singular vectors are good indications of strong/weak input or output directions. Theorem 2.4 Let A ∈ Fm×n . There exist unitary matrices = [u1 , u2 , . . . , um ] ∈ Fm×m U = [v1 , v2 , . . . , vn ] ∈ Fn×n V such that ∗ Σ=  σ1 0 .. . 0 σ2 .. . ··· ··· .. . 0 0 .. . 0 0 ··· σp A = U ΣV , where and    Σ1 =   Σ1 0 0 0  ,      σ1 ≥ σ2 ≥ · · · ≥ σp ≥ 0, p = min{m, n}. Proof. Let σ = kAk and without loss of generality assume m ≥ n. Then, from the definition of kAk, there exists a z ∈ Fn such that kAzk = σ kzk . By Lemma 2.2, there is a matrix Ũ ∈ F m×n such that Ũ ∗ Ũ = I and Az = σŨ z. Now let x= z ∈ Fn , kzk We have Ax = σy. Let V = and U=   y= Ũ z x V1 y Ũ z U1   ∈ Fm . ∈ Fn×n ∈ Fm×m be unitary.2 Consequently, U ∗ AV has the following structure:      ∗ σ σy ∗ y y ∗ AV1 y Ax y ∗ AV1 ∗ = = A1 := U AV = 0 σU1∗ y U1∗ AV1 U1∗ Ax U1∗ AV1 w∗ B  , 2 Recall that it is always possible to extend an orthonormal set of vectors to an orthonormal basis for the whole space. 20 LINEAR ALGEBRA where w := V1∗ A∗ y ∈ Fn−1 and B := U1∗ AV1 ∈ F(m−1)×(n−1) . Since   2 1  0    A∗1  .  = (σ2 + w∗ w),  ..  0 2 2 2 ∗ it follows that kA1 k ≥ σ + w w. But since σ = kAk = kA1 k, we must have w = 0. An obvious induction argument gives U ∗ AV = Σ. This completes the proof. ✷ The σi is the ith singular value of A, and the vectors ui and vj are, respectively, the ith left singular vector and the jth right singular vector. It is easy to verify that Avi A∗ ui = σi ui = σi vi . The preceding equations can also be written as A∗ Avi AA∗ ui = σi2 vi = σi2 ui . Hence σi2 is an eigenvalue of AA∗ or A∗ A, ui is an eigenvector of AA∗ , and vi is an eigenvector of A∗ A. The following notations for singular values are often adopted: σ(A) = σmax (A) = σ1 = the largest singular value of A; and σ(A) = σmin (A) = σp = the smallest singular value of A. Geometrically, the singular values of a matrix A are precisely the lengths of the semiaxes of the hyperellipsoid E defined by E = {y : y = Ax, x ∈ Cn , kxk = 1}. Thus v1 is the direction in which kyk is largest for all kxk = 1; while vn is the direction in which kyk is smallest for all kxk = 1. From the input/output point of view, v1 (vn ) is the highest (lowest) gain input (or control) direction, while u1 (um ) is the highest (lowest) gain output (or observing) direction. This can be illustrated by the following 2 × 2 matrix:     cos θ2 − sin θ2 σ1 cos θ1 − sin θ1 . A= sin θ2 cos θ2 σ2 sin θ1 cos θ1 2.6. Singular Value Decomposition 21 It is easy to see that A maps a unit circle to an ellipsoid with semiaxes of σ1 and σ2 . Hence it is often convenient to introduce the following alternative definitions for the largest singular value σ: σ(A) := max kAxk kxk=1 and for the smallest singular value σ of a tall matrix: σ(A) := min kAxk . kxk=1 Lemma 2.5 Suppose A and ∆ are square matrices. Then (i) |σ(A + ∆) − σ(A)| ≤ σ(∆); (ii) σ(A∆) ≥ σ(A)σ(∆); (iii) σ(A−1 ) = 1 if A is invertible. σ(A) Proof. (i) By definition σ(A + ∆) := ≥ = min k(A + ∆)xk ≥ min {kAxk − k∆xk} kxk=1 kxk=1 min kAxk − max k∆xk kxk=1 kxk=1 σ(A) − σ(∆). Hence −σ(∆) ≤ σ(A + ∆) − σ(A). The other inequality σ(A + ∆) − σ(A) ≤ σ(∆) follows by replacing A by A + ∆ and ∆ by −∆ in the preceding proof. (ii) This follows by noting that σ(A∆) := = ≥ min kA∆xk r min x∗ ∆∗ A∗ A∆x kxk=1 kxk=1 σ(A) min k∆xk = σ(A)σ(∆). kxk=1 (iii) Let the singular value decomposition of A be A = U ΣV ∗ ; then A−1 = V Σ−1 U ∗ . Hence σ(A−1 ) = σ(Σ−1 ) = 1/σ(Σ) = 1/σ(A). ✷ Note that (ii) may not be true if A and ∆ are not square matrices. For example, √   1 consider A = and ∆ = 3 4 ; then σ(A∆) = 0 but σ(A) = 5 and σ(∆) = 5. 2 22 LINEAR ALGEBRA Some useful properties of SVD are collected in the following lemma. Lemma 2.6 Let A ∈ Fm×n and σ1 ≥ σ2 ≥ · · · ≥ σr > σr+1 = · · · = 0, r ≤ min{m, n}. Then 1. rank(A) = r; 2. KerA = span{vr+1 , . . . , vn } and (KerA)⊥ = span{v1 , . . . , vr }; 3. ImA = span{u1 , . . . , ur } and (ImA)⊥ = span{ur+1 , . . . , um }; 4. A ∈ Fm×n has a dyadic expansion: A= r X σi ui vi∗ = Ur Σr Vr∗ , i=1 where Ur = [u1 , . . . , ur ], Vr = [v1 , . . . , vr ], and Σr = diag (σ1 , . . . , σr ); 5. kAk2F = σ12 + σ22 + · · · + σr2 ; 6. kAk = σ1 ; 7. σi (U0 AV0 ) = σi (A), i = 1, . . . , p for any appropriately dimensioned unitary matrices U0 and V0 ; Pk 8. Let k < r = rank(A) and Ak := i=1 σi ui vi∗ ; then min kA − Bk = kA − Ak k = σk+1 . rank(B)≤k Proof. We shall only give a proof for part 8. It is easy to see that rank(Ak ) ≤ k and kA − Ak k = σk+1 . Hence, we only need show that min kA − Bk ≥ σk+1 . Let B rank(B)≤k be any matrix such that rank(B) ≤ k. Then = kU ΣV ∗ − Bk = kΣ − U ∗ BV k     Ik+1 ∗ Ik+1 0 (Σ − U BV ) ≥ = Σk+1 − B̂ , 0     ∗ Ik+1 ∈ F(k+1)×(k+1) and rank(B̂) ≤ k. Let x ∈ Fk+1 where B̂ = Ik+1 0 U BV 0 be such that B̂x = 0 and kxk = 1. Then kA − Bk kA − Bk ≥ Σk+1 − B̂ ≥ (Σk+1 − B̂)x = kΣk+1 xk ≥ σk+1 . Since B is arbitrary, the conclusion follows. ✷ 2.7. Semidefinite Matrices 23 Illustrative MATLAB Commands: ≫ [U, Σ, V] = svd(A) % A = U ΣV ∗ Related MATLAB Commands: cond, condest 2.7 Semidefinite Matrices A square Hermitian matrix A = A∗ is said to be positive definite (semidefinite), denoted by A > 0 (≥ 0), if x∗ Ax > 0 (≥ 0) for all x 6= 0. Suppose A ∈ Fn×n and A = A∗ ≥ 0; then there exists a B ∈ Fn×r with r ≥ rank(A) such that A = BB ∗ . Lemma 2.7 Let B ∈ Fm×n and C ∈ Fk×n . Suppose m ≥ k and B ∗ B = C ∗ C. Then there exists a matrix U ∈ Fm×k such that U ∗ U = I and B = U C. Proof. Let V1 and V2 be unitary matrices such that     B1 C1 B = V1 , C = V2 , 0 0 where B1 and C1 are full-row rank. Then B1 and C1 have the same number of rows and V3 := B1 C1∗ (C1 C1∗ )−1 satisfies V3∗ V3 = I since B ∗ B = C ∗ C. Hence V3 is a unitary matrix and V3∗ B1 = C1 . Finally, let   V3 0 V2∗ U = V1 0 V4 for any suitably dimensioned V4 such that V4∗ V4 = I. by ✷ We can define square root for a positive semidefinite matrix A, A1/2 = (A1/2 )∗ ≥ 0, A = A1/2 A1/2 . Clearly, A1/2 can be computed by using spectral decomposition or SVD: Let A = U ΛU ∗ ; then A1/2 = U Λ1/2 U ∗ , where Λ = diag{λ1 , . . . , λn }, Λ1/2 = diag{ p p λ1 , . . . , λn }. 24 LINEAR ALGEBRA Lemma 2.8 Suppose A = A∗ > 0 and B = B ∗ ≥ 0. Then A > B iff ρ(BA−1 ) < 1. Proof. Since A > 0, we have A > B iff 0 < I − A−1/2 BA−1/2 i.e., iff ρ(A−1/2 BA−1/2 ) < 1. However, A−1/2 BA−1/2 and BA−1 are similar, hence ρ(BA−1 ) = ρ(A−1/2 BA−1/2 ) and the claim follows. ✷ 2.8 Notes and References A very extensive treatment of most topics in this chapter can be found in Brogan [1991], Horn and Johnson [1990, 1991] and Lancaster and Tismenetsky [1985]. Golub and Van Loan’s book [1983] contains many numerical algorithms for solving most of the problems in this chapter. 2.9 Problems Problem 2.1 Let  1 1  A = 2  1 2 1 0 1 0 0  0 1  1.  1 2 Determine the row and column rank of A and find bases for Im(A), Im(A∗ ), and Ker(A).   1 4 Problem 2.2 Let D0 =  2 5 . Find a D such that D∗ D = I and ImD = ImD0 . 3 6   Furthermore, find a D⊥ such that D D⊥ is a unitary matrix. Problem 2.3 Let A be a nonsingular matrix and x, y ∈ Cn . Show (A−1 + xy ∗ )−1 = A − and det(A−1 + xy ∗ )−1 = Axy ∗ A 1 + y ∗ Ax det A . 1 + y ∗ Ax Problem 2.4 Let A and B be compatible matrices. Show B(I + AB)−1 = (I + BA)−1 B, (I + A)−1 = I − A(I + A)−1 . 2.9. Problems 25 Problem 2.5 Find a basis for the maximum dimensional stable invariant subspace of  A R H= with −Q −A∗     −1 2 −1 −1 1. A = , R= , and Q = 0 3 0 −1 −1       0 1 0 0 1 2 2. A = , R= , and Q = 0 2 0 −1 2 4   1 2 3. A = 0, R = , and Q = I2 . 2 5 Problem 2.6 Let A = [aij ]. Show that α(A) := maxi,j |aij | defines a matrix norm. Give examples so that α(A) < ρ(A) and α(AB) > α(A)α(B).     1 2 3 0 Problem 2.7 Let A = and B = . (a) Find all x such that Ax = B. 4 1 −1 1 (b) Find the minimal norm solution x: min {kxk : Ax = B}.     1 2 3 Problem 2.8 Let A =  −2 −5  and B =  4 . Find an x such that kAx − Bk 0 1 5 is minimized. Problem 2.9 Let kAk < 1. Show 1. (I − A)−1 = I + A + A2 + · · · . 2 2. (I − A)−1 ≤ 1 + kAk + kAk + · · · = 3. (I − A)−1 ≥ 1 1−kAk . 1 1+kAk . Problem 2.10 Let A ∈ Cm×n . Show that √ 1 √ kAk2 ≤ kAk∞ ≤ n kAk2 ; m √ 1 √ kAk2 ≤ kAk1 ≤ m kAk2 ; n 1 kAk∞ ≤ kAk1 ≤ m kAk∞ . n Problem 2.11 Let A = xy ∗ and x, y ∈ Cn . Show that kAk2 = kAkF = kxk kyk.   1 1 1+j Problem 2.12 Let A = . Find A 2 and a B ∈ C2 such that A = BB ∗ . 1−j 2 26 LINEAR ALGEBRA Problem 2.13 Let P = P ∗ = λi (P11 ), ∀ 1 ≤ i ≤ k.  P11 ∗ P12 P12 P22  ≥ 0 with P11 ∈ Ck×k . Show λi (P ) ≥  X11 X12 Problem 2.14 Let X = X ≥ 0 be partitioned as X = . (a) Show ∗ X12 X22 ∗ KerX22 ⊂ KerX12 ; (b) let X22 = U2 diag(Λ1 , 0)U2 be such that Λ1 is nonsingular and + + ∗ define X22 := U2 diag(Λ−1 1 , 0)U2 (the pseudoinverse of X22 ); then show that Y = X12 X22 solves Y X22 = X12 ; and (c) show that       + + ∗ I 0 X11 X12 I X12 X22 X11 − X12 X22 X12 0 = . + ∗ ∗ X12 X22 X22 X12 0 I 0 X22 I ∗  Chapter 3 Linear Systems This chapter reviews some basic system theoretical concepts. The notions of controllability, observability, stabilizability, and detectability are defined and various algebraic and geometric characterizations of these notions are summarized. Observer theory is then introduced. System interconnections and realizations are studied. Finally, the concepts of system poles and zeros are introduced. 3.1 Descriptions of Linear Dynamical Systems Let a finite dimensional linear time invariant (FDLTI) dynamical system be described by the following linear constant coefficient differential equations: ẋ = Ax + Bu, x(t0 ) = x0 y = Cx + Du, (3.1) (3.2) where x(t) ∈ Rn is called the system state, x(t0 ) is called the initial condition of the system, u(t) ∈ Rm is called the system input, and y(t) ∈ Rp is the system output. The A, B, C, and D are appropriately dimensioned real constant matrices. A dynamical system with single-input (m = 1) and single-output (p = 1) is called a SISO (singleinput and single-output) system; otherwise it is called a MIMO (multiple-input and multiple-output) system. The corresponding transfer matrix from u to y is defined as Y (s) = G(s)U (s), where U (s) and Y (s) are the Laplace transforms of u(t) and y(t) with zero initial condition (x(0) = 0). Hence, we have G(s) = C(sI − A)−1 B + D. Note that the system equations (3.1) and (3.2) can be written in a more compact matrix form:      ẋ A B x = . y C D u 27 28 LINEAR SYSTEMS To expedite calculations involving transfer matrices, we shall use the following notation:   A B := C(sI − A)−1 B + D. C D In Matlab the system can also be written in the packed form using the command ≫ G=pck(A, B, C, D) % pack the realization in partitioned form ≫ seesys(G) % display G in partitioned format ≫ [A, B, C, D]=unpck(G) % unpack the system matrix Note that  A C B D  is a real block matrix, not a transfer function. Illustrative MATLAB Commands: ≫ G=pck([], [], [], 10) % create a constant system matrix ≫ [y, x, t]=step(A, B, C, D, Iu) % Iu=i (step response of the ith channel) ≫ [y, x, t]=initial(A, B, C, D, x0 ) % initial response with initial condition x0 ≫ [y, x, t]=impulse(A, B, C, D, Iu) % impulse response of the Iuth channel ≫ [y,x]=lsim(A,B,C,D,U,T) % U is a length(T ) × column(B) matrix input; T is the sampling points. Related MATLAB Commands: minfo, trsp, cos tr, sin tr, siggen 3.2 Controllability and Observability We now turn to some very important concepts in linear system theory. Definition 3.1 The dynamical system described by equation (3.1) or the pair (A, B) is said to be controllable if, for any initial state x(0) = x0 , t1 > 0 and final state x1 , there exists a (piecewise continuous) input u(·) such that the solution of equation (3.1) satisfies x(t1 ) = x1 . Otherwise, the system or the pair (A, B) is said to be uncontrollable. The controllability (and the observability introduced next) of a system can be verified through some algebraic or geometric criteria. 3.2. Controllability and Observability 29 Theorem 3.1 The following are equivalent: (i) (A, B) is controllable. (ii) The matrix Wc (t) := Z t ∗ eAτ BB ∗ eA τ dτ 0 is positive definite for any t > 0. (iii) The controllability matrix  . . . An−1 B Pn has full-row rank or, in other words, hA |ImBi := i=1 Im(Ai−1 B) = Rn . C=  B AB A2 B (iv) The matrix [A − λI, B] has full-row rank for all λ in C. (v) Let λ and x be any eigenvalue and any corresponding left eigenvector of A (i.e., x∗ A = x∗ λ); then x∗ B 6= 0. (vi) The eigenvalues of A+BF can be freely assigned (with the restriction that complex eigenvalues are in conjugate pairs) by a suitable choice of F .         2 0 1 1 0 and B = . Then x1 = and x2 = 0 2 1 0 1 are independent eigenvectors of A and x∗i B 6= 0, i = 1, 2. However, this should not lead one to conclude that (A, B) is controllable. In fact, x = x1 − x2 is also an eigenvector of A and x∗ B = 0, which implies that (A, B) is not controllable. Hence one must check for all possible eigenvectors in using criterion (v). Example 3.1 Let A = Definition 3.2 An unforced dynamical system ẋ = Ax is said to be stable if all the eigenvalues of A are in the open left half plane; that is, Reλ(A) < 0. A matrix A with such a property is said to be stable or Hurwitz. Definition 3.3 The dynamical system of equation (3.1), or the pair (A, B), is said to be stabilizable if there exists a state feedback u = F x such that the system is stable (i.e., A + BF is stable). It is more appropriate to call this stabilizability the state feedback stabilizability to differentiate it from the output feedback stabilizability defined later. The following theorem is a consequence of Theorem 3.1. 30 LINEAR SYSTEMS Theorem 3.2 The following are equivalent: (i) (A, B) is stabilizable. (ii) The matrix [A − λI, B] has full-row rank for all Reλ ≥ 0. (iii) For all λ and x such that x∗ A = x∗ λ and Reλ ≥ 0, x∗ B 6= 0. (iv) There exists a matrix F such that A + BF is Hurwitz. We now consider the dual notions: observability and detectability of the system described by equations (3.1) and (3.2). Definition 3.4 The dynamical system described by equations (3.1) and (3.2) or by the pair (C, A) is said to be observable if, for any t1 > 0, the initial state x(0) = x0 can be determined from the time history of the input u(t) and the output y(t) in the interval of [0, t1 ]. Otherwise, the system, or (C, A), is said to be unobservable. Theorem 3.3 The following are equivalent: (i) (C, A) is observable. (ii) The matrix Wo (t) := Z t ∗ eA τ C ∗ CeAτ dτ 0 is positive definite for any t > 0. (iii) The observability matrix     O=   C CA CA2 .. . CAn−1        T has full-column rank or ni=1 Ker(CAi−1 ) = 0.   A − λI (iv) The matrix has full-column rank for all λ in C. C (v) Let λ and y be any eigenvalue and any corresponding right eigenvector of A (i.e., Ay = λy); then Cy 6= 0. (vi) The eigenvalues of A+LC can be freely assigned (with the restriction that complex eigenvalues are in conjugate pairs) by a suitable choice of L. (vii) (A∗ , C ∗ ) is controllable. 3.3. Observers and Observer-Based Controllers 31 Definition 3.5 The system, or the pair (C, A), is detectable if A + LC is stable for some L. Theorem 3.4 The following are equivalent: (i) (C, A) is detectable.   A − λI (ii) The matrix has full-column rank for all Reλ ≥ 0. C (iii) For all λ and x such that Ax = λx and Reλ ≥ 0, Cx 6= 0. (iv) There exists a matrix L such that A + LC is Hurwitz. (v) (A∗ , C ∗ ) is stabilizable. The conditions (iv) and (v) of Theorems 3.1 and 3.3 and the conditions (ii) and (iii) of Theorems 3.2 and 3.4 are often called Popov-Belevitch-Hautus (PBH) tests. In particular, the following definitions of modal controllability and observability are often useful. Definition 3.6 Let λ be an eigenvalue of A or, equivalently, a mode of the system. Then the mode λ is said to be controllable (observable) if x∗ B 6= 0 (Cx 6= 0) for all left (right) eigenvectors of A associated with λ; that is, x∗ A = λx∗ (Ax = λx) and 0 6= x ∈ Cn . Otherwise, the mode is said to be uncontrollable (unobservable). It follows that a system is controllable (observable) if and only if every mode is controllable (observable). Similarly, a system is stabilizable (detectable) if and only if every unstable mode is controllable (observable). Illustrative MATLAB Commands: ≫ C= ctrb(A, B); O= obsv(A, C); ≫ Wc (∞)=gram(A, B); % if A is stable. ≫ F=-place(A, B, P) % P is a vector of desired eigenvalues. Related MATLAB Commands: ctrbf, obsvf, canon, strans, acker 3.3 Observers and Observer-Based Controllers It is clear that if a system is controllable and the system states are available for feedback, then the system closed-loop poles can be assigned arbitrarily through a constant feedback. However, in most practical applications, the system states are not completely accessible and all the designer knows are the output y and input u. Hence, the estimation of the system states from the given output information y and input u is often 32 LINEAR SYSTEMS necessary to realize some specific design objectives. In this section, we consider such an estimation problem and the application of this state estimation in feedback control. Consider a plant modeled by equations (3.1) and (3.2). An observer is a dynamical system with input (u, y) and output (say, x̂), that asymptotically estimates the state x, that is, x̂(t) − x(t) → 0 as t → ∞ for all initial states and for every input. Theorem 3.5 An observer exists iff (C, A) is detectable. Further, if (C, A) is detectable, then a full-order Luenberger observer is given by q̇ = Aq + Bu + L(Cq + Du − y) x̂ = q, (3.3) (3.4) where L is any matrix such that A + LC is stable. Recall that, for a dynamical system described by the equations (3.1) and (3.2), if (A, B) is controllable and state x is available for feedback, then there is a state feedback u = F x such that the closed-loop poles of the system can be arbitrarily assigned. Similarly, if (C, A) is observable, then the system observer poles can be arbitrarily placed so that the state estimator x̂ can be made to approach x arbitrarily fast. Now let us consider what will happen if the system states are not available for feedback so that the estimated state has to be used. Hence, the controller has the following dynamics: x̂˙ = (A + LC)x̂ + Bu + LDu − Ly u = F x̂. Then the total system state equations are given by      ẋ A BF x = . −LC A + BF + LC x̂ x̂˙ Let e := x − x̂; then the system equation becomes      ė A + LC 0 e = −LC A + BF x̂ x̂˙ and the closed-loop poles consist of two parts: the poles resulting from state feedback λi (A + BF ) and the poles resulting from the state estimation λj (A + LC). Now if (A, B) is controllable and (C, A) is observable, then there exist F and L such that the eigenvalues of A + BF and A + LC can be arbitrarily assigned. In particular, they can be made to be stable. Note that a slightly weaker result can also result even if (A, B) and (C, A) are only stabilizable and detectable. The controller given above is called an observer-based controller and is denoted as u = K(s)y and K(s) =  A + BF + LC + LDF F −L 0  . 3.3. Observers and Observer-Based Controllers 33 Now denote the open-loop plant by G=  B D A C  ; then the closed-loop feedback system is as shown below: y ✛ G ✛ u K ✛ In general, if a system is stabilizable through feeding back the output y, then it is said to be output feedback stabilizable. It is clear from the above construction that a system is output feedback stabilizable if and only if (A, B) is stabilizable and (C, A) is detectable.  1 Example 3.2 Let A = 1 state feedback u = F x such done by choosing F = −6      2 1 ,B = , and C = 1 0 . We shall design a 0 0 that the closed-loop poles are at {−2, −3}. This can be −8 using ≫ F = −place(A, B, [−2, −3]). Now suppose the states are not available for feedback and we want to construct an  −21 observer so that the observer poles are at {−10, −10}. Then L = can be −51 obtained by using ≫ L = −acker(A′ , C ′ , [−10, −10])′ and the observer-based controller is given by K(s) = −534(s + 0.6966) . (s + 34.6564)(s − 8.6564) Note that the stabilizing controller itself is unstable. Of course, this may not be desirable in practice. 34 3.4 LINEAR SYSTEMS Operations on Systems In this section, we present some facts about system interconnection. Since these proofs are straightforward, we will leave the details to the reader. Suppose that G1 and G2 are two subsystems with state-space representations:     A2 B2 A1 B1 , G2 = . G1 = C1 D1 C2 D2 Then the series or cascade connection of these two subsystems is a system with the output of the second subsystem as the input of the first subsystem, as shown in the following diagram: ✛ G1 ✛ G2 ✛ This operation in terms of the transfer matrices of the two subsystems is essentially the product of two transfer matrices. Hence, a representation for the cascaded system can be obtained as    A2 B2 A1 B1 G1 G2 = C1 D1 C2 D2     A1 B1 C2 A2 0 B1 D2 B2  =  B1 C2 A1 A2 B2 B1 D2  . =  0 C1 D1 C2 D1 C2 C1 D1 D2 D1 D2 Similarly, the parallel connection or the addition of G1 and G2 can be obtained as       B1 A1 0 A1 B1 A2 B2 . B2 G1 + G2 = + =  0 A2 C1 D1 C2 D2 C1 C2 D1 + D2 For future reference, we shall also introduce the following definitions. Definition 3.7 The transpose of a transfer matrix G(s) or the dual system is defined as G 7−→ GT (s) = B ∗ (sI − A∗ )−1 C ∗ + D∗ or, equivalently,  A C B D  7−→  A∗ B∗ C∗ D∗  . Definition 3.8 The conjugate system of G(s) is defined as G 7−→ G∼ (s) := GT (−s) = B ∗ (−sI − A∗ )−1 C ∗ + D∗ or, equivalently,  A C B D  7−→  −A∗ B∗ −C ∗ D∗  . 3.5. State-Space Realizations for Transfer Matrices 35 ∗ In particular, we have G∗ (jω) := [G(jω)] = G∼ (jω). A real rational matrix Ĝ(s) is called an inverse of a transfer matrix G(s) if G(s)Ĝ(s) = Ĝ(s)G(s) = I. Suppose G(s) is square and D is invertible. Then   A − BD−1 C −BD−1 . G−1 = D−1 C D−1 The corresponding Matlab commands are G1 G2 ⇐⇒ mmult(G1 , G2 ),   G1 G2  ⇐⇒ sbs(G1 , G2 ) G1 + G2 ⇐⇒ madd(G1 , G2 ), G1 − G2 ⇐⇒ msub(G1 , G2 )    G1 G1 ⇐⇒ abv(G1 , G2 ), ⇐⇒ daug(G1 , G2 ), G2 G2 GT (s) ⇐⇒ transp(G), G∼ (s) ⇐⇒ cjt(G), G−1 (s) ⇐⇒ minv(G) α G(s) ⇐⇒ mscl(G, α), α is a scalar. Related MATLAB Commands: append, parallel, feedback, series, cloop, sclin, sclout, sel 3.5 State-Space Realizations for Transfer Matrices In some cases, the natural or convenient description for a dynamical system is in terms of matrix transfer function. This occurs, for example, in some highly complex systems for which the analytic differential equations are too hard or too complex to write down. Hence certain engineering approximation or identification has to be carried out; for example, input and output frequency responses are obtained from experiments so that some transfer matrix approximating the system dynamics can be obtained. Since the state-space computation is most convenient to implement on the computer, some appropriate state-space representation for the resulting transfer matrix is necessary. In general, assume that G(s) is a real rational transfer matrix that is proper. Then we call a state-space model (A, B, C, D) such that   A B G(s) = C D a realization of G(s). Definition 3.9 A state-space realization (A, B, C, D) of G(s) is said to be a minimal realization of G(s) if A has the smallest possible dimension. Theorem 3.6 A state-space realization (A, B, C, D) of G(s) is minimal if and only if (A, B) is controllable and (C, A) is observable. 36 LINEAR SYSTEMS We now describe several ways to obtain a state-space realization for a given multipleinput and multiple-output transfer matrix G(s). We shall first consider SIMO (singleinput and multiple-output) and MISO (multiple-input and single-output) systems. Let G(s) be a column vector of transfer function with p outputs: β1 sn−1 + β2 sn−2 + · · · + βn−1 s + βn sn + a1 sn−1 + · · · + an−1 s + an   A b with Then G(s) = C d  −a1 −a2 · · · −an−1 −an  1 0 ··· 0 0   0 1 · · · 0 0 A :=   .. .. .. ..  . . . . G(s) = 0 C= 0  ··· β1 β2 1 0 · · · βn−1 + d, βi ∈ Rp , d ∈ Rp .        βn      b :=     1 0 0 .. .       0 is a so-called controllable canonical form or controller canonical form. Dually, consider a multiple-input and single-output system G(s) = η1 sn−1 + η2 sn−2 + · · · + ηn−1 s + ηn + d, ηi∗ ∈ Rm , d∗ ∈ Rm . sn + a1 sn−1 + · · · + an−1 s + an Then G(s) =  A B c d   −a1 −a2 .. .     =  −an−1   −an 1 1 0 ··· 0 1 ··· .. .. . . 0 0 ··· 0 0 ··· 0 0 0 .. . η1 η2 .. . 1 ηn−1 ηn 0 0 ··· 0 d          is an observable canonical form or observer canonical form. For a MIMO system, the simplest and most straightforward way to obtain a realization is by realizing each element of the matrix G(s) and then combining all these individual realizations to form a realization for G(s). To illustrate, let us consider a 2 × 2 (block) transfer matrix such as   G1 (s) G2 (s) G(s) = G3 (s) G4 (s) and assume that Gi (s) has a state-space realization of the form   Ai Bi Gi (s) = , i = 1, . . . , 4. Ci Di 3.5. State-Space Realizations for Transfer Matrices 37 Then a realization for G(s) can be obtained as (G = abv(sbs(G1 , G2 ), sbs(G3 , G4 ))):     G(s) =     A1 0 0 0 C1 0 0 A2 0 0 C2 0 0 0 A3 0 0 C3 0 0 0 A4 0 C4 B1 0 B3 0 D1 D3 0 B2 0 B4 D2 D4     .    Alternatively, if the transfer matrix G(s) can be factored into the product and/or the sum of several simply realized transfer matrices, then a realization for G can be obtained by using the cascade or addition formulas given in the preceding section. A problem inherited with these kinds of realization procedures is that a realization thus obtained will generally not be minimal. To obtain a minimal realization, a Kalman controllability and observability decomposition has to be performed to eliminate the uncontrollable and/or unobservable states. (An alternative numerically reliable method to eliminate uncontrollable and/or unobservable states is the balanced realization method, which will be discussed later.) We shall now describe a procedure that does result in a minimal realization by using partial fractional expansion (the resulting realization is sometimes called Gilbert’s realization due to E. G. Gilbert). Let G(s) be a p × m transfer matrix and write it in the following form: G(s) = N (s) d(s) with d(s) a scalar polynomial. For simplicity, we shall assume that d(s) has only real and distinct roots λi 6= λj if i 6= j and d(s) = (s − λ1 )(s − λ2 ) · · · (s − λr ). Then G(s) has the following partial fractional expansion: G(s) = D + r X Wi . s − λi i=1 Suppose rank Wi = ki and let Bi ∈ Rki ×m and Ci ∈ Rp×ki be two constant matrices such that Wi = Ci Bi . 38 LINEAR SYSTEMS Then a realization for G(s) is given by  λ1 Ik1   G(s) =   C1 .. . ··· λr Ikr Cr  B1 ..  .  . Br  D It follows immediately from PBH tests that this realization is controllable and observable, and thus it is minimal. An immediate consequence of this minimal realization is that a transfer matrix with an rth order polynomial denominator does not necessarily have an rth order state-space realization unless Wi for each i is a rank one matrix. This approach can, in fact, be generalized to more complicated cases where d(s) may have complex and/or repeated roots. Readers may convince themselves by trying some simple examples. Illustrative MATLAB Commands: ≫ G=nd2sys(num, den, gain); G=zp2sys(zeros, poles, gain); Related MATLAB Commands: ss2tf, ss2zp, zp2ss, tf2ss, residue, minreal 3.6 Multivariable System Poles and Zeros A matrix is called a polynomial matrix of a variable if every element of the matrix is a polynomial of the variable. Definition 3.10 Let Q(s) be a (p × m) polynomial matrix (or a transfer matrix) of s. Then the normal rank of Q(s), denoted normalrank (Q(s)), is the maximally possible rank of Q(s) for at least one s ∈ C. To show the difference between the normal rank of a polynomial matrix and the rank of the polynomial matrix evaluated at a certain point, consider   s 1 Q(s) =  s2 1  . s 1 Then Q(s) has normal rank 2 since rank Q(3) = 2. However, Q(0) has rank 1. The poles and zeros of a transfer matrix can be characterized in terms of its statespace realizations. Let   A B C D be a state-space realization of G(s). 3.6. Multivariable System Poles and Zeros 39 Definition 3.11 The eigenvalues of A are called the poles of the realization of G(s). To define zeros, let us consider the following system matrix:   A − sI B Q(s) = . C D Definition 3.12 A complex number z0 ∈ C is called an invariant zero of the system realization if it satisfies     A − z0 I B A − sI B rank < normalrank . C D C D The invariant zeros are not changed by constant state feedback since      A + BF − z0 I B A − z0 I B I 0 rank = rank C + DF D C D F I   A − z0 I B = rank . C D It is also clear that invariant zeros are not changed under similarity transformation. The following lemma is obvious.   A − sI B Lemma 3.7 Suppose has full-column normal rank. Then z0 ∈ C is an C D invariant zero of a realization (A, B, C, D) if and only if there exist 0 6= x ∈ Cn and u ∈ Cm such that    A − z0 I B x = 0. C D u Moreover, if u = 0, then z0 is also a nonobservable mode. Proof. By definition, z0 is an invariant zero if there is a vector  since  A − sI C B D  A − z0 I C B D  x u   x u  6= 0 such that =0 has full-column normal rank. On the other hand, suppose z0 is an invariant zero; then there is a vector such that  A − z0 I C B D  x u  = 0.  x u  6= 0 40 LINEAR SYSTEMS    B A − sI We claim that x 6= 0. Otherwise, u = 0 or u = 0 since D C   x full-column normal rank (i.e., = 0), which is a contradiction. u Finally, note that if u = 0, then   A − z0 I x=0 C B D  and z0 is a nonobservable mode by PBH test. has ✷ When the system is square, the invariant zeros can be computed by solving a generalized eigenvalue problem:       A B I 0 x x = z0 C D 0 0 u u | {z } | {z } M N using a Matlab command: eig(M, N).   A − sI B Lemma 3.8 Suppose has full-row normal rank. Then z0 ∈ C is an C D invariant zero of a realization (A, B, C, D) if and only if there exist 0 6= y ∈ Cn and v ∈ Cp such that    ∗  A − z0 I B y v∗ = 0. C D Moreover, if v = 0, then z0 is also a noncontrollable mode. Lemma 3.9 G(s) has full-column (row) normal rank if and only if full-column (row) normal rank. Proof. This follows by noting that    A − sI B I = C D C(A − sI)−1 and normalrank  A − sI C B D  0 I  A − sI 0  A − sI C B G(s) B D  has  = n + normalrank(G(s)). ✷ Lemma 3.10 Let G(s) ∈ Rp (s) be a p × m transfer matrix and let (A, B, C, D) be a minimal realization. If the system input is of the form u(t) = u0 eλt , where λ ∈ C is not a pole of G(s) and u0 ∈ Cm is an arbitrary constant vector, then the output due to the input u(t) and the initial state x(0) = (λI − A)−1 Bu0 is y(t) = G(λ)u0 eλt , ∀t ≥ 0. In particular, if λ is a zero of G(s), then y(t) = 0. 3.7. Notes and References 41 Proof. The system response with respect to the input u(t) = u0 eλt and the initial condition x(0) = (λI − A)−1 Bu0 is (in terms of the Laplace transform) Y (s) = C(sI − A)−1 x(0) + C(sI − A)−1 BU (s) + DU (s) = C(sI − A)−1 x(0) + C(sI − A)−1 Bu0 (s − λ)−1 + Du0 (s − λ)−1   = C(sI − A)−1 x(0) + C (sI − A)−1 − (λI − A)−1 Bu0 (s − λ)−1 +C(λI − A)−1 Bu0 (s − λ)−1 + Du0 (s − λ)−1 = C(sI − A)−1 (x(0) − (λI − A)−1 Bu0 ) + G(λ)u0 (s − λ)−1 = G(λ)u0 (s − λ)−1 . Hence y(t) = G(λ)u0 eλt . ✷ Example 3.3 Let G(s) =  A C B D     =    −1 −2 1 1 2 3 0 2 −1 3 2 1   −4 −3 −2 1 1 1  . 1 1 1 0 0 0  2 3 4 0 0 0 Then the invariant zeros of the system can be found using the Matlab command ≫ G=pck(A, B, C, D), z0 = szeros(G), % or ≫ z0 = tzero(A, B, C, D) which gives z0 = 0.2. Since G(s) is full-row rank, we can find y and v such that    ∗  A − z0 I B y v∗ = 0, C D which can again be computed using a Matlab command:  0.0466    0.0466  y ≫ null([A − z0 ∗ eye(3), B; C, D]′ ) =⇒ =  −0.1866 v  −0.9702 0.1399    .   Related MATLAB Commands: spoles, rifd 3.7 Notes and References Readers are referred to Brogan [1991], Chen [1984], Kailath [1980], and Wonham [1985] for extensive treatment of standard linear system theory. 42 3.8 LINEAR SYSTEMS Problems Problem 3.1 Let A ∈ Cn×n and B ∈ Cm×m . Show that X(t) = eAt X(0)eBt is the solution to Ẋ = AX + XB. Problem 3.2 Given the impulse response, h1 (t), of a linear time-invariant system with model ÿ + a1 ẏ + a2 y = u find the impulse response, h(t), of the system ÿ + a1 ẏ + a2 y = b0 ü + b1 u̇ + b2 u. Justify your answer and generalize your result to nth order systems. Problem 3.3 a second order by ẋ = Ax. Suppose  Suppose   system is given    it is known 4 1 5 1 that x(1) = for x(0) = and x(1) = for x(0) = . Find x(n) −2 1 −2 2   1 for x(0) = . Can you determine A? 0 Problem 3.4 Assume (A, B) is controllable. Show that (F, G) with     A 0 B F = , G= C 0 0 is controllable if and only if  A C B 0  is a full-row rank matrix. Problem 3.5 Let λi , i = 1, 2, . . . , n be n distinct eigenvalues of a matrix A ∈ Cn×n and let xi and yi be the corresponding right- and left-unit eigenvectors such that yi∗ xi = 1. Show that n X λi xi yi∗ yi∗ xj = δij , A = i=1 and −1 C(sI − A) B= n X (Cxi )(y ∗ B) i i=1 s − λi . Furthermore, show that the mode λi is controllable iff yi∗ B 6= 0 and the mode λi is observable iff Cxi 6= 0. 3.8. Problems 43 Problem 3.6 Let (A, b, c) be a realization with A ∈ Rn×n , b ∈ Rn , and c∗ ∈ Rn . Assume that λi (A) + λj (A) 6= 0 for all i, j. This assumption ensures the existence of X ∈ Rn×n such that AX + XA + bc = 0 Show that X is nonsingular if and only if (A, b) is controllable and (c, A) is observable. Problem 3.7 Compute the system zeros and the corresponding zero following transfer functions    −1 −2 1 2 1 2 2 2  0  1 0 3 4  1 2 1   G1 (s) =   0 1 1 2  , G2 (s) =  1 1 0 0 1 0 2 0 1 1 0 0 G3 (s) =  2(s + 1)(s + 2) s(s + 3)(s + 4) s+2 (s + 1)(s + 3)  ,  1  s+1 G4 (s) =   10 directions of the   ,   s+3 (s + 1)(s − 2)    5 s−2 s+3 Also find the vectors x and u whenever appropriate so that either       ∗  A − zI B A − zI B x ∗ x u = 0 or = 0. C D C D u 44 LINEAR SYSTEMS Chapter 4 H2 and H∞ Spaces The most important objective of a control system is to achieve certain performance specifications in addition to providing internal stability. One way to describe the performance specifications of a control system is in terms of the size of certain signals of interest. For example, the performance of a tracking system could be measured by the size of the tracking error signal. In this chapter, we look at several ways of defining a signal’s size (i.e., at several norms for signals). Of course, which norm is most appropriate depends on the situation at hand. For that purpose, we shall first introduce the Hardy spaces H2 and H∞ . Some state-space methods of computing real rational H2 and H∞ transfer matrix norms are also presented. 4.1 Hilbert Spaces Recall the inner product of vectors defined on a Euclidean space    x1 y1 n X  ..   .. ∗ x̄i yi ∀x =  .  , y =  . hx, yi := x y = i=1 xn yn Cn :   n ∈C . Note that many important metric notions and geometrical properties, such as length, distance, angle, and the energy of physical systems, can be deduced from this inner product. For instance, the length of a vector x ∈ Cn is defined as p kxk := hx, xi and the angle between two vectors x, y ∈ Cn can be computed from cos ∠(x, y) = hx, yi , ∠(x, y) ∈ [0, π]. kxk kyk The two vectors are said to be orthogonal if ∠(x, y) = π2 . 45 46 H2 AND H∞ SPACES We now consider a natural generalization of the inner product on Cn to more general (possibly infinite dimensional) vector spaces. Definition 4.1 Let V be a vector space over C. An inner product1 on V is a complexvalued function, h·, ·i : V × V 7−→ C such that for any x, y, z ∈ V and α, β ∈ C (i) hx, αy + βzi = αhx, yi + βhx, zi (ii) hx, yi = hy, xi (iii) hx, xi > 0 if x 6= 0. A vector space V with an inner product is called an inner product space. p It is clear that the inner product defined above induces a norm kxk := hx, xi, so that the norm conditions in Chapter 2 are satisfied. In particular, the distance between vectors x and y is d(x, y) = kx − yk. Two vectors x and y in an inner product space V are said to be orthogonal if hx, yi = 0, denoted x ⊥ y. More generally, a vector x is said to be orthogonal to a set S ⊂ V , denoted by x ⊥ S, if x ⊥ y for all y ∈ S. The inner product and the inner product induced norm have the following familiar properties. Theorem 4.1 Let V be an inner product space and let x, y ∈ V . Then (i) |hx, yi| ≤ kxk kyk (Cauchy-Schwarz inequality). Moreover, the equality holds if and only if x = αy for some constant α or y = 0. (ii) kx + yk2 + kx − yk2 = 2 kxk2 + 2 kyk2 (Parallelogram law). 2 2 2 (iii) kx + yk = kxk + kyk if x ⊥ y. A Hilbert space is a complete inner product space with the norm induced by its inner product. For example, Cn with the usual inner product is a (finite dimensional) Hilbert space. More generally, it is straightforward to verify that Cn×m with the inner product defined as m n X X āij bij ∀A, B ∈ Cn×m hA, Bi := trace A∗ B = i=1 j=1 is also a (finite dimensional) Hilbert space. A well-know infinite dimensional Hilbert space is L2 [a, b], which consists of all square integrable and Lebesgue measurable functions defined on an interval [a, b] with the inner product defined as Z b hf, gi := f (t)∗ g(t)dt a 1 The property (i) in the following list is the other way around to the usual mathematical convention since we want to have hx, yi = x∗ y rather than y ∗ x for x, y ∈ Cn . 4.2. H2 and H∞ Spaces 47 for f, g ∈ L2 [a, b]. Similarly, if the functions are vector or matrix-valued, the inner product is defined correspondingly as hf, gi := Z b trace [f (t)∗ g(t)] dt. a Some spaces used often in this book are L2 [0, ∞), L2 (−∞, 0], L2 (−∞, ∞). More precisely, they are defined as L2 = L2 (−∞, ∞): Hilbert space of matrix-valued functions on R, with inner product Z ∞ hf, gi := trace [f (t)∗ g(t)] dt. −∞ L2+ = L2 [0, ∞): subspace of L2 (−∞, ∞) with functions zero for t < 0. L2− = L2 (−∞, 0]: subspace of L2 (−∞, ∞) with functions zero for t > 0. 4.2 H2 and H∞ Spaces Let S ⊂ C be an open set, and let f (s) be a complex-valued function defined on S: f (s) : S 7−→ C. Then f (s) is said to be analytic at a point z0 in S if it is differentiable at z0 and also at each point in some neighborhood of z0 . It is a fact that if f (s) is analytic at z0 then f has continuous derivatives of all orders at z0 . Hence, a function analytic at z0 has a power series representation at z0 . The converse is also true (i.e., if a function has a power series at z0 , then it is analytic at z0 ). A function f (s) is said to be analytic in S if it has a derivative or is analytic at each point of S. A matrix-valued function is analytic in S if every element of the matrix is analytic in S. For example, all real rational stable transfer matrices are analytic in the right-half plane and e−s is analytic everywhere. A well-know property of the analytic functions is the so-called maximum modulus theorem. Theorem 4.2 If f (s) is defined and continuous on a closed-bounded set S and analytic on the interior of S, then |f (s)| cannot attain the maximum in the interior of S unless f (s) is a constant. The theorem implies that |f (s)| can only achieve its maximum on the boundary of S; that is, max |f (s)| = max |f (s)| s∈S s∈∂S where ∂S denotes the boundary of S. Next we consider some frequently used complex (matrix) function spaces. 48 H2 AND H∞ SPACES L2 (jR) Space L2 (jR) or simply L2 is a Hilbert space of matrix-valued (or scalar-valued) functions on jR and consists of all complex matrix functions F such that the following integral is bounded: Z ∞ trace [F ∗ (jω)F (jω)] dω < ∞. −∞ The inner product for this Hilbert space is defined as Z ∞ 1 trace [F ∗ (jω)G(jω)] dω hF, Gi := 2π −∞ for F, G ∈ L2 , and the inner product induced norm is given by p kF k2 := hF, F i. For example, all real rational strictly proper transfer matrices with no poles on the imaginary axis form a subspace (not closed) of L2 (jR) that is denoted by RL2 (jR) or simply RL2 . H2 Space2 H2 is a (closed) subspace of L2 (jR) with matrix functions F (s) analytic in Re(s) > 0 (open right-half plane). The corresponding norm is defined as   Z ∞ 1 2 kF k2 := sup trace [F ∗ (σ + jω)F (σ + jω)] dω . σ>0 2π −∞ It can be shown3 that kF k22 = 1 2π Z ∞ trace [F ∗ (jω)F (jω)] dω. −∞ Hence, we can compute the norm for H2 just as we do for L2 . The real rational subspace of H2 , which consists of all strictly proper and real rational stable transfer matrices, is denoted by RH2 . H2⊥ Space H2⊥ is the orthogonal complement of H2 in L2 ; that is, the (closed) subspace of functions in L2 that are analytic in the open left-half plane. The real rational subspace of H2⊥ , which consists of all strictly proper rational transfer matrices with all poles in the open right-half plane, will be denoted by RH⊥ 2 . It is easy to see that if G is a strictly proper, stable, and real rational transfer matrix, then G ∈ H2 and G∼ ∈ H2⊥ . Most of our study in this book will be focused on the real rational case. 2 The H space and H ∞ space defined in this subsection together with the Hp spaces, p ≥ 1, 2 which will not be introduced in this book, are usually called Hardy spaces and are named after the mathematician G. H. Hardy (hence the notation of H). 3 See Francis [1987]. 4.2. H2 and H∞ Spaces 49 The L2 spaces defined previously in the frequency domain can be related to the L2 spaces defined in the time domain. Recall the fact that a function in L2 space in the time domain admits a bilateral Laplace (or Fourier) transform. In fact, it can be shown that this bilateral Laplace transform yields an isometric isomorphism between the L2 spaces in the time domain and the L2 spaces in the frequency domain (this is what is called Parseval’s relations): L2 (−∞, ∞) ∼ = L2 (jR) L2 [0, ∞) ∼ = H2 L2 (−∞, 0] ∼ = H2⊥ . As a result, if g(t) ∈ L2 (−∞, ∞) and if its bilateral Laplace transform is G(s) ∈ L2 (jR), then kGk2 = kgk2 . Hence, whenever there is no confusion, the notation for functions in the time domain and in the frequency domain will be used interchangeably. L2 [0, ∞) ✛ Laplace Transform ✲ Inverse Transform ✻ H2 ✻ P+ P+ L2 (−∞, ∞) P− ✛ Laplace Transform ✲ Inverse Transform P− ❄ L2 (−∞, 0] L2 (jR) ❄ ✛ Laplace Transform ✲ Inverse Transform H2⊥ Figure 4.1: Relationships among function spaces Define an orthogonal projection P+ : L2 (−∞, ∞) 7−→ L2 [0, ∞) such that, for any function f (t) ∈ L2 (−∞, ∞), we have g(t) = P+ f (t) with  f (t), for t ≥ 0; g(t) := 0, for t < 0. 50 H2 AND H∞ SPACES In this book, P+ will also be used to denote the projection from L2 (jR) onto H2 . Similarly, define P− as another orthogonal projection from L2 (−∞, ∞) onto L2 (−∞, 0] (or L2 (jR) onto H2⊥ ). Then the relationships between L2 spaces and H2 spaces can be shown as in Figure 4.1. Other classes of important complex matrix functions used in this book are those bounded on the imaginary axis. L∞ (jR) Space L∞ (jR) or simply L∞ is a Banach space of matrix-valued (or scalar-valued) functions that are (essentially) bounded on jR, with norm kF k∞ := ess sup σ [F (jω)] . ω∈R The rational subspace of L∞ , denoted by RL∞ (jR) or simply RL∞ , consists of all proper and real rational transfer matrices with no poles on the imaginary axis. H∞ Space H∞ is a (closed) subspace of L∞ with functions that are analytic and bounded in the open right-half plane. The H∞ norm is defined as kF k∞ := sup σ [F (s)] = sup σ [F (jω)] . Re(s)>0 ω∈R The second equality can be regarded as a generalization of the maximum modulus theorem for matrix functions. (See Boyd and Desoer [1985] for a proof.) The real rational subspace of H∞ is denoted by RH∞ , which consists of all proper and real rational stable transfer matrices. − H∞ Space − H∞ is a (closed) subspace of L∞ with functions that are analytic and bounded in − the open left-half plane. The H∞ norm is defined as kF k∞ := sup σ [F (s)] = sup σ [F (jω)] . Re(s)<0 ω∈R − The real rational subspace of H∞ is denoted by RH− ∞ , which consists of all proper, real rational, antistable transfer matrices (i.e., functions with all poles in the open right-half plane). Let G(s) ∈ L∞ be a p × q transfer matrix. Then a multiplication operator is defined as MG : L2 7−→ L2 MG f := Gf. 4.2. H2 and H∞ Spaces 51 In writing the preceding mapping, we have assumed that f has a compatible dimension. A more accurate description of the foregoing operator should be MG : Lq2 7−→ Lp2 . That is, f is a q-dimensional vector function with each component in L2 . However, we shall suppress all dimensions in this book and assume that all objects have compatible dimensions. A useful fact about the multiplication operator is that the norm of a matrix G in L∞ equals the norm of the corresponding multiplication operator. Theorem 4.3 Let G ∈ L∞ be a p × q transfer matrix. Then kMG k = kGk∞ . Remark 4.1 It is also true that this operator norm equals the norm of the operator restricted to H2 (or H2⊥ ); that is, kMG k = kMG |H2 k := sup {kGf k2 : f ∈ H2 , kf k2 ≤ 1} . This will be clear in the proof where an f ∈ H2 is constructed. ✸ Proof. By definition, we have kMG k = sup {kGf k2 : f ∈ L2 , kf k2 ≤ 1} . First we see that kGk∞ is an upper bound for the operator norm: Z ∞ 1 2 f ∗ (jω)G∗ (jω)G(jω)f (jω) dω kGf k2 = 2π −∞ Z ∞ 1 kf (jω)k2 dω ≤ kGk2∞ 2π −∞ = kGk2∞ kf k22 . To show that kGk∞ is the least upper bound, first choose a frequency ω0 where σ [G(jω)] is maximum; that is, σ [G(jω0 )] = kGk∞ , and denote the singular value decomposition of G(jω0 ) by G(jω0 ) = σu1 (jω0 )v1∗ (jω0 ) + r X σi ui (jω0 )vi∗ (jω0 ) i=2 where r is the rank of G(jω0 ) and ui , vi have unit length. Next we assume that G(s) has real coefficients and we shall construct a function f (s) ∈ H2 with real coefficients so that the norm is approximately achieved. [It will be clear in the following that the proof is much simpler if f is allowed to have complex coefficients, which is necessary when G(s) has complex coefficients.] 52 H2 AND H∞ SPACES If ω0 < ∞, write v1 (jω0 ) as    v1 (jω0 ) =   α1 ejθ1 α2 ejθ2 .. . αq ejθq      where αi ∈ R is such that θi ∈ (−π, 0] and q is the column dimension of G. Now let 0 ≤ βi ≤ ∞ be such that   βi − jω0 θi = ∠ βi + jω0 (with βi → ∞ if θi = 0) and let f be given by    f (s) =    (with 1 replacing βi −s βi +s α1 ββ11 −s +s −s α2 ββ22 +s .. . βq −s αq βq +s    ˆ  f(s)   if θi = 0), where a scalar function fˆ is chosen so that ˆ |f(jω)| =  c if |ω − ω0 | < ǫ or |ω + ω0 | < ǫ 0 otherwise ˆ where p ǫ is a small positive number and c is chosen so that f has unit 2-norm (i.e., c = π/2ǫ). This, in turn, implies that f has unit 2-norm. Then kGf k22 i 1 h 2 2 σ [G(−jω0 )] π + σ [G(jω0 )] π 2π = σ [G(jω0 )]2 = kGk2∞ . ≈ Similarly, if ω0 = ∞, the conclusion follows by letting ω0 → ∞ in the foregoing. ✷ Illustrative MATLAB Commands: ≫ [sv, w]=sigma(A, B, C, D); % frequency response of the singular values; or ≫ w=logspace(l, h, n); sv=sigma(A, B, C, D, w); % n points between 10l and 10h . Related MATLAB Commands: semilogx, semilogy, bode, freqs, nichols, frsp, vsvd, vplot, pkvnorm 4.3. Computing L2 and H2 Norms 4.3 53 Computing L2 and H2 Norms Let G(s) ∈ L2 and recall that the L2 norm of G is defined as s Z ∞ 1 trace{G∗ (jω)G(jω)} dω kGk2 := 2π −∞ = = kgk sZ2 ∞ trace{g ∗ (t)g(t)} dt −∞ where g(t) denotes the convolution kernel of G. It is easy to see that the L2 norm defined previously is finite iff the transfer matrix G is strictly proper; that is, G(∞) = 0. Hence, we will generally assume that the transfer matrix is strictly proper whenever we refer to the L2 norm of G (of course, this also applies to H2 norms). One straightforward way of computing the L2 norm is to use contour integral. Suppose G is strictly proper; then we have Z ∞ 1 2 kGk2 = trace{G∗ (jω)G(jω)} dω 2π −∞ I 1 trace{G∼ (s)G(s)} ds. = 2πj The last integral is a contour integral along the imaginary axis and around an infinite semicircle in the left-half plane; the contribution to the integral from this semicircle equals zero because G is strictly proper. By the residue theorem, kGk22 equals the sum of the residues of trace{G∼ (s)G(s)} at its poles in the left-half plane. Although kGk2 can, in principle, be computed from its definition or from the method just suggested, it is useful in many applications to have alternative characterizations and to take advantage of the state-space representations of G. The computation of a RH2 transfer matrix norm is particularly simple. Lemma 4.4 Consider a transfer matrix G(s) =  A C B 0  with A stable. Then we have kGk22 = trace(B ∗ QB) = trace(CP C ∗ ) (4.1) where Q and P are observability and controllability Gramians that can be obtained from the following Lyapunov equations: AP + P A∗ + BB ∗ = 0 A∗ Q + QA + C ∗ C = 0. 54 H2 AND H∞ SPACES Proof. Since G is stable, we have −1 g(t) = L  (G) = CeAt B, 0, t≥0 t<0 and kGk22 Z ∞ trace{g ∗ (t)g(t)} dt = Z ∞ trace{g(t)g(t)∗ } dt 0 Z ∞ Z0 ∞ ∗ ∗ A∗ t ∗ At trace{CeAt BB ∗ eA t C ∗ } dt. trace{B e C Ce B} dt = = = 0 0 The lemma follows from the fact that the controllability Gramian of (A, B) and the observability Gramian of (C, A) can be represented as Z ∞ Z ∞ ∗ ∗ Q= eA t C ∗ CeAt dt, P = eAt BB ∗ eA t dt, 0 0 which can also be obtained from AP + P A∗ + BB ∗ = 0 A∗ Q + QA + C ∗ C = 0. ✷ To compute the L2 norm of a rational transfer function, G(s) ∈ RL2 , using the state-space approach, let G(s) = [G(s)]+ + [G(s)]− with G+ ∈ RH2 and G− ∈ RH⊥ 2; then 2 2 2 kGk2 = k[G(s)]+ k2 + k[G(s)]− k2 ∼ where k[G(s)]+ k2 and k[G(s)]− k2 = k[G(−s)]+ k2 = k([G(s)]− ) k2 can be computed using the preceding lemma. Still another useful characterization of the H2 norm of G is in terms of hypothetical input-output experiments. Let ei denote the ith standard basis vector of Rm , where m is the input dimension of the system. Apply the impulsive input δ(t)ei [δ(t) is the unit impulse] and denote the output by zi (t)(= g(t)ei ). Assume D = 0; then zi ∈ L2+ and kGk22 = m X i=1 kzi k22 . Note that this characterization of the H2 norm can be appropriately generalized for nonlinear time-varying systems; see Chen and Francis [1992] for an application of this norm in sampled-data control. Example 4.1 Consider a transfer matrix  3(s + 3)  (s − 1)(s + 2) G= s+1 (s + 2)(s + 3)  2 s−1 =G +G s u 1  s−4 4.4. Computing L∞ and H∞ Norms with 55  −2 0 −1 0  0 −3 2 0  , Gs =   1 0 0 0  1 1 0 0   1  0 Gu =   1 0 0 4 0 1 4 0 0 0  2 1  . 0  0 Then the command h2norm(G qs ) gives kGs k2 = 0.6055 and h2norm(cjt(Gu )) gives 2 2 kGu k2 = 3.182. Hence kGk2 = kGs k2 + kGu k2 = 3.2393. Illustrative MATLAB Commands: ≫ P = gram(A, B); Q = gram(A′ , C′ ); or P = lyap(A, B ∗ B′ ); ≫ [Gs , Gu ] = sdecomp(G); % decompose into stable and antistable parts. 4.4 Computing L∞ and H∞ Norms We shall first consider, as in the L2 case, how to compute the ∞ norm of an RL∞ transfer matrix. Let G(s) ∈ RL∞ and recall that the L∞ norm of a matrix rational transfer function G is defined as kGk∞ := sup σ{G(jω)}. ω The computation of the L∞ norm of G is complicated and requires a search. A control engineering interpretation of the infinity norm of a scalar transfer function G is the distance in the complex plane from the origin to the farthest point on the Nyquist plot of G, and it also appears as the peak value on the Bode magnitude plot of |G(jω)|. Hence the ∞ norm of a transfer function can, in principle, be obtained graphically. To get an estimate, set up a fine grid of frequency points: {ω1 , · · · , ωN }. Then an estimate for kGk∞ is max σ{G(jωk )}. 1≤k≤N This value is usually read directly from a Bode singular value plot. The RL∞ norm can also be computed in state-space. Lemma 4.5 Let γ > 0 and G(s) =  A C B D  ∈ RL∞ . (4.2) 56 H2 AND H∞ SPACES Then kGk∞ < γ if and only if σ(D) < γ and the Hamiltonian matrix H has no eigenvalues on the imaginary axis where   A + BR−1 D∗ C BR−1 B ∗ H := (4.3) −C ∗ (I + DR−1 D∗ )C −(A + BR−1 D∗ C)∗ and R = γ 2 I − D∗ D. Proof. Let Φ(s) = γ 2 I − G∼ (s)G(s). Then it is clear that kGk∞ < γ if and only if Φ(jω) > 0 for all ω ∈ R. Since Φ(∞) = R > 0 and since Φ(jω) is a continuous function of ω, Φ(jω) > 0 for all ω ∈ R if and only if Φ(jω) is nonsingular for all ω ∈ R ∪ {∞}; that is, Φ(s) has no imaginary axis zero. Equivalently, Φ−1 (s) has no imaginary axis pole. It is easy to compute by some simple algebra that     BR−1 H   −C ∗ DR−1 Φ−1 (s) =  .  −1 ∗  R D C R−1 B ∗ R−1 Thus the conclusion follows if the above realization has neither uncontrollable modes nor unobservable modes on the imaginary axis. Assume that jω0 is an eigenvalue of H but not a pole of Φ−1 (s). Then jω0 must be either anunobservable mode of   BR−1 ( R−1 D∗ C R−1 B ∗ , H) or an uncontrollable mode of (H, ). Now −C ∗ DR−1  −1 ∗  −1 ∗ suppose jω , H). Then there exists  0 is an unobservable mode of ( R D C R B x1 6= 0 such that an x0 = x2 Hx0 = jω0 x0 ,  R−1 D∗ C R−1 B ∗ These equations can be simplified to (jω0 I − A)x1 (jω0 I + A∗ )x2 D∗ Cx1 + B ∗ x2  x0 = 0. = 0 = −C ∗ Cx1 = 0. Since A has no imaginary axis eigenvalues, we have x1 = 0 and x2 = 0. This contradicts our assumption, and hence the realization has no unobservable modes on the imaginary axis. Similarly, a contradiction will also   be arrived at if jω0 is assumed to be an unconBR−1 trollable mode of (H, ). ✷ −C ∗ DR−1 4.4. Computing L∞ and H∞ Norms 57 Bisection Algorithm Lemma 4.5 suggests the following bisection algorithm to compute RL∞ norm: (a) Select an upper bound γu and a lower bound γl such that γl ≤ kGk∞ ≤ γu ; (b) If (γu − γl )/γl ≤specified level, stop; kGk ≈ (γu + γl )/2. Otherwise go to the next step; (c) Set γ = (γl + γu )/2; (d) Test if kGk∞ < γ by calculating the eigenvalues of H for the given γ; (e) If H has an eigenvalue on jR, set γl = γ; otherwise set γu = γ; go back to step (b). Of course, the above algorithm applies to H∞ norm computation as well. Thus L∞ norm computation requires a search, over either γ or ω, in contrast to L2 (H2 ) norm computation, which does not. A somewhat analogous situation occurs for constant matrices with the norms kM k22 = trace(M ∗ M ) and kM k∞ = σ[M ]. In principle, kM k22 can be computed exactly with a finite number of operations, as can the test for whether σ(M ) < γ (e.g., γ 2 I − M ∗ M > 0), but the value of σ(M ) cannot. To compute σ(M ), we must use some type of iterative algorithm. Remark 4.2 It is clear that kGk∞ < γ iff γ −1 G ∞ < 1. Hence, there is no loss of generality in assuming γ = 1. This assumption will often be made in the remainder of this book. It is also noted that there are other fast algorithms to carry out the preceding norm computation; nevertheless, this bisection algorithm is the simplest. ✸ Additional interpretations can be given for the H∞ norm of a stable matrix transfer function. When G(s) is a single-input and single-output system, the H∞ norm of the G(s) can be regarded as the largest possible amplification factor of the system’s steadystate response to sinusoidal excitations. For example, the steady-state response of the system with respect to a sinusoidal input u(t) = U sin(ω0 t + φ) is y(t) = U |G(jω0 )| sin (ω0 t + φ + ∠G(jω0 )) and thus the maximum possible amplification factor is sup |G(jω0 )|, which is precisely ω0 the H∞ norm of the transfer function. In the multiple-input and multiple-output case, the H∞ norm of a transfer matrix G ∈ RH∞ can also be regarded as the largest possible amplification factor of the system’s steady-state response to sinusoidal excitations in the following sense: Let the sinusoidal inputs be     u1 sin(ω0 t + φ1 ) u1  u2 sin(ω0 t + φ2 )   u2      u(t) =   , û =  ..  . ..    .  . uq sin(ω0 t + φq ) uq 58 H2 AND H∞ SPACES Then the steady-state response of the system can be written as    y(t) =   y1 sin(ω0 t + θ1 ) y2 sin(ω0 t + θ2 ) .. . yp sin(ω0 t + θp )    ,     ŷ =   y1 y2 .. . yp      for some yi , θi , i = 1, 2, . . . , p, and furthermore, kŷk φi ,ωo ,û kûk kGk∞ = sup where k·k is the Euclidean norm. The details are left as an exercise. Example 4.2 Consider a mass/spring/damper system as shown in Figure 4.2. F1 x1 m1 k1 F2 b1 x2 m2 k2 b2 Figure 4.2: A two-mass/spring/damper system The dynamical system can be described by the following differential equations:   ẋ1   ẋ2      ẋ3  = A  ẋ4   x1   x2   + B F1 x3  F2 x4 4.4. Computing L∞ and H∞ Norms 59 2 10 The largest singular value 1 10 0 10 −1 The smallest singular value 10 −2 10 −1 0 10 1 10 10 frequency (rad/sec) Figure 4.3: kGk∞ is the peak of the largest singular value of G(jω) with     A=   0 0 k1 − m1 k1 m2 0 0 k1 m1 k1 + k2 − m2 1 0 b1 − m1 b1 m2 0 1 b1 m1 b1 + b2 − m2         , B =      0 0 1 m1 0 0 0 0 1 m2     .   Suppose that G(s) is the transfer matrix from (F1 , F2 ) to (x1 , x2 ); that is,   1 0 0 0 C= , D = 0, 0 1 0 0 and suppose k1 = 1, k2 = 4, b1 = 0.2, b2 = 0.1, m1 = 1, and m2 = 2 with appropriate units. The following Matlab commands generate the singular value Bode plot of the above system as shown in Figure 4.3. ≫ G=pck(A,B,C,D); ≫ hinfnorm(G,0.0001) or linfnorm(G,0.0001) % relative error ≤ 0.0001 ≫ w=logspace(-1,1,200); % 200 points between 1 = 10−1 and 10 = 101; ≫ Gf=frsp(G,w); % computing frequency response; ≫ [u,s,v]=vsvd(Gf ); % SVD at each frequency; 60 H2 AND H∞ SPACES ≫ vplot(′ liv, lm′ , s), grid % plot both singular values and grid. Then the H∞ norm of this transfer matrix is kG(s)k∞ = 11.47, which is shown as the peak of the largest singular value Bode plot in Figure 4.3. Since the peak is achieved at ωmax = 0.8483, exciting the system using the following sinusoidal input     0.9614 sin(0.8483t) F1 = 0.2753 sin(0.8483t − 0.12) F2 gives the steady-state response of the system as     11.47 × 0.9614 sin(0.8483t − 1.5483) x1 = . 11.47 × 0.2753 sin(0.8483t − 1.4283) x2 This shows that the system response will be amplified 11.47 times for an input signal at the frequency ωmax , which could be undesirable if F1 and F2 are disturbance force and x1 and x2 are the positions to be kept steady. Example 4.3 Consider a two-by-two transfer matrix  10(s + 1) 1  s2 + 0.2s + 100 s + 1 G(s) =   s+2 5(s + 1) s2 + 0.1s + 10 (s + 2)(s + 3)   .  A state-space realization of G can be obtained using the following Matlab commands: ≫ G11=nd2sys([10,10],[1,0.2,100]); ≫ G12=nd2sys(1,[1,1]); ≫ G21=nd2sys([1,2],[1,0.1,10]); ≫ G22=nd2sys([5,5],[1,5,6]); ≫ G=sbs(abv(G11,G21),abv(G12,G22)); Next, we set up a frequency grid to compute the frequency response of G and the singular values of G(jω) over a suitable range of frequency. ≫ w=logspace(0,2,200); % 200 points between 1 = 100 and 100 = 102 ; ≫ Gf=frsp(G,w); % computing frequency response; ≫ [u,s,v]=vsvd(Gf ); % SVD at each frequency; 4.5. Notes and References 61 ≫ vplot(′ liv, lm′ , s), grid % plot both singular values and grid; ≫ pkvnorm(s) % find the norm from the frequency response of the singular values. The singular values of G(jω) are plotted in Figure 4.4, which gives an estimate of kGk∞ ≈ 32.861. The state-space bisection algorithm described previously leads to kGk∞ = 50.25 ± 0.01 and the corresponding Matlab command is ≫ hinfnorm(G,0.0001) or linfnorm(G,0.0001) % relative error ≤ 0.0001. 2 10 1 10 0 10 −1 10 −2 10 0 10 1 10 2 10 Figure 4.4: The largest and the smallest singular values of G(jω) The preceding computational results show clearly that the graphical method can lead to a wrong answer for a lightly damped system if the frequency grid is not sufficiently dense. Indeed, we would get kGk∞ ≈ 43.525, 48.286 and 49.737 from the graphical method if 400, 800, and 1600 frequency points are used, respectively. Related MATLAB Commands: linfnorm, vnorm, getiv, scliv, var2con, xtract, xtracti 4.5 Notes and References The basic concept of function spaces presented in this chapter can be found in any standard functional analysis textbook, for instance, Naylor and Sell [1982] and Gohberg and Goldberg [1981]. The system theoretical interpretations of the norms and function 62 H2 AND H∞ SPACES spaces can be found in Desoer and Vidyasagar [1975]. The bisection L∞ norm computational algorithm was first developed in Boyd, Balakrishnan, and Kabamba [1989]. A more efficient L∞ norm computational algorithm is presented in Bruinsma and Steinbuch [1990]. 4.6 Problems Problem 4.1 Let G(s) be a matrix in RH∞ . Prove that   2 G = kGk2∞ + 1. I ∞ Problem 4.2 (Parseval relation) Let f (t), g(t) ∈ L2 , F (jω) = F{f (t)}, and G(jω) = F{g(t)}. Show that Z ∞ Z ∞ 1 F (jω)G∗ (jω)dω f (t)g(t)dt = 2π −∞ −∞ and Z ∞ −∞ Note that F (jω) = Z ∞ |f (t)|2 dt = f (t)e−jωt dt, −∞ where F −1 1 2π Z ∞ −∞ |F (jω)|2 dω. f (t) = F −1 (F (jω)) = 1 2π Z ∞ F (jω)ejωt dω. −∞ denotes the inverse Fourier transform. Problem 4.3 Suppose A is stable. Show Z ∞ (jωI − A)−1 dω = πI.   −∞ A B ∈ RH∞ and let Q = Q∗ be the observability Gramian. Use C 0 the above formula to show that Z ∞ 1 G∼ (jω)G(jω)dω = B ∗ QB. 2π −∞ Suppose G(s) = [Hint: Use the fact that G∼ (s)G(s) = F ∼ (s) + F (s) and F (s) = B ∗ Q(sI − A)−1 B.] Problem 4.4 Compute the 2-norm and ∞-norm of the following   1 s+3  1  s + 1 (s + 1)(s − 2)   , G2 (s) =  2 G1 (s) =    10 5 1 s−2 s+3 systems:  0 1 3 1  2 0 4.6. Problems 63   −1 −2 1 0 0 , G3 (s) =  1 2 3 0    G4 (s) =     −1 −2 −3 1 2 1 0 0 0 1   0 1 0 2 0  . 1 0 0 1 0  0 1 1 0 2 Problem 4.5 Let r(t) = sin ωt be the input signal to a plant G(s) = s2 ωn2 + 2ξωn s + ωn2 √ with 0 < ξ < 1/ 2. Find the steady-state response of the system y(t). Also find the frequency ω that gives the largest magnitude steady-state response of y(t). Problem 4.6 Let G(s) ∈ RH∞ be a p × q transfer matrix and y = G(s)u. Suppose     u1 u1 sin(ω0 t + φ1 )  u2   u2 sin(ω0 t + φ2 )      u(t) =   , û =  ..  . ..   .   . uq sin(ω0 t + φq ) uq Show that the steady-state response of the system is given by    y1 y1 sin(ω0 t + θ1 )  y2  y2 sin(ω0 t + θ2 )     y(t) =   , ŷ =  .. ..   .  . yp yp sin(ω0 t + θp ) for some yi and θi , i = 1, 2, . . . , p. Show that sup φi ,ωo ,kûk2 ≤1      kŷk2 = kGk∞ . Problem 4.7 Write a Matlab program to plot, versus γ, the distance from the imaginary axis to the nearest eigenvalue of the Hamiltonian matrix for a given state-space model with stable A. Try it on   s+1 s  (s + 2)(s + 3)    s2 − 2 (s + 3)(s + 4) s+1 s+4 (s + 1)(s + 2)   .  Read off the value of the H∞ -norm. Compare with the Matlab function hinfnorm. 1 . Compute kG(s)k∞ using the Bode (s2 + 2ξs + 1)(s + 1) plot and state-space algorithm, respectively for ξ = 1, 0.1, 0.01, 0.001 and compare the results. Problem 4.8 Let G(s) = 64 H2 AND H∞ SPACES Chapter 5 Internal Stability This chapter introduces the feedback structure and discusses its stability and various stability tests. The arrangement of this chapter is as follows: Section 5.1 discusses the necessity for introducing feedback structure and describes the general feedback configuration. Section 5.2 defines the well-posedness of the feedback loop. Section 5.3 introduces the notion of internal stability and various stability tests. Section 5.4 introduces the stable coprime factorizations of rational matrices. The stability conditions in terms of various coprime factorizations are also considered in this section. 5.1 Feedback Structure In designing control systems, there are several fundamental issues that transcend the boundaries of specific applications. Although they may differ for each application and may have different levels of importance, these issues are generic in their relationship to control design objectives and procedures. Central to these issues is the requirement to provide satisfactory performance in the face of modeling errors, system variations, and uncertainty. Indeed, this requirement was the original motivation for the development of feedback systems. Feedback is only required when system performance cannot be achieved because of uncertainty in system characteristics. A more detailed treatment of model uncertainties and their representations will be discussed in Chapter 8. For the moment, assuming we are given a model including a representation of uncertainty that we believe adequately captures the essential features of the plant, the next step in the controller design process is to determine what structure is necessary to achieve the desired performance. Prefiltering input signals (or open-loop control) can change the dynamic response of the model set but cannot reduce the effect of uncertainty. If the uncertainty is too great to achieve the desired accuracy of response, then a feedback structure is required. The mere assumption of a feedback structure, however, does not guarantee a reduction of uncertainty, and there are many obstacles to achieving the uncertainty-reducing benefits of feedback. In particular, since for any 65 66 INTERNAL STABILITY reasonable model set representing a physical system uncertainty becomes large and the phase is completely unknown at sufficiently high frequencies, the loop gain must be small at those frequencies to avoid destabilizing the high-frequency system dynamics. Even worse is that the feedback system actually increases uncertainty and sensitivity in the frequency ranges where uncertainty is significantly large. In other words, because of the type of sets required to model physical systems reasonably and because of the restriction that our controllers be causal, we cannot use feedback (or any other control structure) to cause our closed-loop model set to be a proper subset of the open-loop model set. Often, what can be achieved with intelligent use of feedback is a significant reduction of uncertainty for certain signals of importance with a small increase spread over other signals. Thus, the feedback design problem centers around the tradeoff involved in reducing the overall impact of uncertainty. This tradeoff also occurs, for example, when using feedback to reduce command/disturbance error while minimizing response degradation due to measurement noise. To be of practical value, a design technique must provide means for performing these tradeoffs. We shall discuss these tradeoffs in more detail in the next chapter. To focus our discussion, we shall consider the standard feedback configuration shown in Figure 5.1. It consists of the interconnected plant P and controller K forced by command r, sensor noise n, plant input disturbance di , and plant output disturbance d. In general, all signals are assumed to be multivariable, and all transfer matrices are assumed to have appropriate dimensions. r✲e − ✻ ✲ K u di u ❄ ✲ e p✲ P d ❄ ✲ e y✲ n ❄ e✛ Figure 5.1: Standard feedback configuration 5.2 Well-Posedness of Feedback Loop Assume that the plant P and the controller K in Figure 5.1 are fixed real rational proper transfer matrices. Then the first question one would ask is whether the feedback interconnection makes sense or is physically realizable. To be more specific, consider a simple example where s−1 , K=1 P =− s+2 5.2. Well-Posedness of Feedback Loop 67 are both proper transfer functions. However, u= s−1 (s + 2) (r − n − d) + di . 3 3 That is, the transfer functions from the external signals r − n − d and di to u are not proper. Hence, the feedback system is not physically realizable. Definition 5.1 A feedback system is said to be well-posed if all closed-loop transfer matrices are well-defined and proper. Now suppose that all the external signals r, n, d, and di are specified and that the closed-loop transfer matrices from them to u are, respectively, well-defined and proper. Then y and all other signals are also well-defined and the related transfer matrices are proper. Furthermore, since the transfer matrices from d and n to u are the same and differ from the transfer matrix from r to u by only a sign, the system is well-posed if di and only if the transfer matrix from to u exists and is proper. d To be consistent with the notation used in the rest of this book, we shall denote K̂ := −K (5.1) and regroup the external input signals into the feedback loop as w1 and w2 and regroup the input signals of the plant and the controller as e1 and e2 . Then the feedback loop with the plant and the controller can be simply represented as inFigure 5.2 and the w1 system is well-posed if and only if the transfer matrix from to e1 exists and is w2 proper. w1 e1 ✲e + ✻ + ✲ P K̂ ✛ e2 + w2 ❄ ✛+ e Figure 5.2: Internal stability analysis diagram Lemma 5.1 The feedback system in Figure 5.2 is well-posed if and only if I − K̂(∞)P (∞) is invertible. (5.2) 68 INTERNAL STABILITY Proof. The system in Figure 5.2 can be represented in equation form as e1 e2 = w1 + K̂e2 = w2 + P e1 . Then an expression for e1 can be obtained as (I − K̂P )e1 = w1 + K̂w2 . Thus well-posedness is equivalent to the condition that (I − K̂P )−1 exists and is proper. But this is equivalent to the condition that the constant term of the transfer function I − K̂P is invertible. ✷ It is straightforward to show that equation (5.2) is equivalent to either one of the following two conditions:   I −K̂(∞) is invertible; (5.3) −P (∞) I I − P (∞)K̂(∞) is invertible. The well-posedness condition is simple to state in terms of state-space realizations. Introduce realizations of P and K̂: # "    B̂ A B . P = , K̂ = C D Ĉ D̂ Then P (∞) = D, K̂(∞) = D̂ and condition in equation (5.3) is  the well-posedness  I −D̂ equivalent to the invertibility of . Fortunately, in most practical cases −D I we shall have D = 0, and hence well-posedness for most practical control systems is guaranteed. 5.3 Internal Stability Consider a system described by the standard block diagram in Figure 5.2 and assume that the system is well-posed. Definition 5.2 The system of Figure 5.2 is said to be internally stable if the transfer matrix  −1   (I − K̂P )−1 K̂(I − P K̂)−1 I −K̂ = (5.4) −P I P (I − K̂P )−1 (I − P K̂)−1   I + K̂(I − P K̂)−1 P K̂(I − P K̂)−1 = (I − P K̂)−1 P (I − P K̂)−1 from (w1 , w2 ) to (e1 , e2 ) belongs to RH∞ . 5.3. Internal Stability 69 Note that to check internal stability, it is necessary (and sufficient) to test whether each of the four transfer matrices in equation (5.4) is in H∞ . Stability cannot be concluded even if three of the four transfer matrices in equation (5.4) are in H∞ . For example, let an interconnected system transfer function be given by P = s−1 , s+1 K̂ = − 1 . s−1 Then it is easy to compute  e1 e2   s+1  s+2 =  s−1 s+2 −  s+1   (s − 1)(s + 2)  w1   w2 , s+1 s+2 which shows that the system is not internally stable although three of the four transfer functions are stable. Remark 5.1 Internal stability is a basic requirement for a practical feedback system. This is because all interconnected systems may be unavoidably subject to some nonzero initial conditions and some (possibly small) errors, and it cannot be tolerated in practice that such errors at some locations will lead to unbounded signals at some other locations in the closed-loop system. Internal stability guarantees that all signals in a system are bounded provided that the injected signals (at any locations) are bounded. ✸ However, there are some special cases under which determining system stability is simple. Corollary 5.2 Suppose K̂ ∈ RH∞ . Then the system in Figure 5.2 is internally stable if and only if it is well-posed and P (I − K̂P )−1 ∈ RH∞ . Proof. The necessity is obvious. To prove the sufficiency, it is sufficient to show that (I − P K̂)−1 ∈ RH∞ . But this follows from (I − P K̂)−1 = I + (I − P K̂)−1 P K̂ and (I − P K̂)−1 P, K̂ ∈ RH∞ . ✷ Also, we have the following: Corollary 5.3 Suppose P ∈ RH∞ . Then the system in Figure 5.2 is internally stable if and only if it is well-posed and K̂(I − P K̂)−1 ∈ RH∞ . Corollary 5.4 Suppose P ∈ RH∞ and K̂ ∈ RH∞ . Then the system in Figure 5.2 is internally stable if and only if (I − P K̂)−1 ∈ RH∞ , or, equivalently, det(I − P (s)K̂(s)) has no zeros in the closed right-half plane. 70 INTERNAL STABILITY Note that all the previous discussions and conclusions apply equally to infinite dimensional plants and controllers. To study the more general case, we shall limit our discussions to finite dimensional systems and define nk np := number of open right-half plane (rhp) poles of K̂(s) := number of open right-half plane (rhp) poles of P (s). Theorem 5.5 The system is internally stable if and only if it is well-posed and (i) the number of open rhp poles of P (s)K̂(s) = nk + np ; (ii) (I − P (s)K̂(s))−1 is stable. Proof. It is easy to show that P K̂ and (I − P K̂)−1 have the following realizations:   " # A B Ĉ B D̂ Ā B̄   0 −1 B̂  , (I − P K̂) =  P K̂ =  C̄ D̄ C DĈ DD̂ where Ā B̄ C̄ D̄      A B Ĉ B D̂ = + (I − DD̂)−1 C 0  B̂   B D̂ = (I − DD̂)−1 B̂   = (I − DD̂)−1 C DĈ DĈ  = (I − DD̂)−1 . Hence, the system is internally stable iff Ā is stable. (see Problem 5.2.) Now suppose that the system is internally stable; then (I − P K̂)−1 ∈ RH∞ . So we only need to show that given condition (ii), condition (i) is necessary and sufficient for the internal stability. This follows by noting that (Ā, B̄) is stabilizable iff     A B Ĉ B D̂ , (5.5) 0  B̂ is stabilizable; and (C̄, Ā) is detectable iff      A B Ĉ C DĈ , 0  is detectable. But conditions (5.5) and (5.6) are equivalent to condition (i). (5.6) ✷ Condition (i) in the preceding theorem implies that there is no unstable pole/zero cancellation in forming the product P K̂. 5.4. Coprime Factorization over RH∞ 71 The preceding theorem is, in fact, the basis for the classical control theory, where the stability is checked only for one closed-loop transfer function with the implicit assumption that the controller itself is stable (and most probably also minimum phase; or at least marginally stable and minimum phase with the condition that any imaginary axis pole of the controller is not in the same location as any zero of the plant). Example 5.1 Let P and K̂ be two-by-two transfer matrices 1  s−1 P = 0  0 1 s+1   ,  1−s K̂ =  s + 1 0 −1 −1  . Then  −1  s+1 P K̂ =   0  −1 s−1  , −1  s+1 (I − P K̂)−1  s+1  s+2 =  0 −  (s + 1)2 (s + 2)2 (s − 1)  .  s+1 s+2 So the closed-loop system is not stable even though det(I − P K̂) = (s + 2)2 (s + 1)2 has no zero in the closed right-half plane and the number of unstable poles of P K̂ = nk + np = 1. Hence, in general, det(I − P K̂) having no zeros in the closed right-half plane does not necessarily imply (I − P K̂)−1 ∈ RH∞ . 5.4 Coprime Factorization over RH∞ Recall that two polynomials m(s) and n(s), with, for example, real coefficients, are said to be coprime if their greatest common divisor is 1 (equivalent, they have no common zeros). It follows from Euclid’s algorithm1 that two polynomials m and n are coprime iff there exist polynomials x(s) and y(s) such that xm + yn = 1; such an equation is called a Bezout identity. Similarly, two transfer functions m(s) and n(s) in RH∞ are said to be coprime over RH∞ if there exists x, y ∈ RH∞ such that xm + yn = 1. 1 See, for example, Kailath [1980], pages 140–141. 72 INTERNAL STABILITY The more primitive, but equivalent, definition is that m and n are coprime if every common divisor of m and n is invertible in RH∞ ; that is, h, mh−1 , nh−1 ∈ RH∞ =⇒ h−1 ∈ RH∞ . More generally, we have the following: Definition 5.3 Two matrices M and N in RH∞ are right coprime over RH∞ if they have the same number of columns and if there exist matrices Xr and Yr in RH∞ such that    M  Xr Yr = Xr M + Yr N = I. N Similarly, two matrices M̃ and Ñ in RH∞ are left coprime over RH∞ if they have the same number of rows and if there exist matrices Xl and Yl in RH∞ such that     Xl = M̃ Xl + ÑYl = I. M̃ Ñ Yl   M Note that these definitions are equivalent to saying that the matrix is left inN   vertible in RH∞ and the matrix M̃ Ñ is right invertible in RH∞ . These two equations are often called Bezout identities. Now let P be a proper real rational matrix. A right coprime factorization (rcf ) of P is a factorization P = N M −1 , where N and M are right coprime over RH∞ . Similarly, a left coprime factorization (lcf ) has the form P = M̃ −1 Ñ , where Ñ and M̃ are left-coprime over RH∞ . A matrix P (s) ∈ Rp (s) is said to have double coprime factorization if there exist a right coprime factorization P = N M −1 , a left coprime factorization P = M̃ −1 Ñ , and Xr , Yr , Xl , Yl ∈ RH∞ such that    Xr Yr M −Yl = I. (5.7) N Xl −Ñ M̃ Of course, implicit in these definitions is the requirement that both M and M̃ be square and nonsingular. Theorem 5.6 Suppose P (s) is a proper real rational matrix and   A B P = C D is a stabilizable and detectable realization. Let F and L be such that A+BF and A+LC are both stable, and define     A + BF B −L M −Yl = (5.8) F I 0  N Xl C + DF D I 5.4. Coprime Factorization over RH∞  Xr −Ñ Yr M̃   A + LC = F C 73  −(B + LD) L I 0 . −D I (5.9) Then P = N M −1 = M̃ −1 Ñ are rcf and lcf, respectively, and, furthermore, equation (5.7) is satisfied. Proof. The theorem follows by verifying equation (5.7). ✷ Remark 5.2 Note that if P is stable, then we can take Xr = Xl = I, Yr = Yl = 0, N = Ñ = P , M = M̃ = I. ✸ Remark 5.3 The coprime factorization of a transfer matrix can be given a feedbackcontrol interpretation. For example, right coprime factorization comes out naturally from changing the control variable by a state feedback. Consider the state-space equations for a plant P : ẋ = Ax + Bu y = Cx + Du. Next, introduce a state feedback and change the variable v := u − F x where F is such that A + BF is stable. Then we get ẋ = (A + BF )x + Bv u = Fx + v y = (C + DF )x + Dv. Evidently, from these equations, the transfer matrix from v to u is   A + BF B M (s) = F I and that from v to y is N (s) =  A + BF C + DF B D  Therefore, u = M v, y = N v so that y = N M −1 u; that is, P = N M −1 . ✸ 74 INTERNAL STABILITY We shall now see how coprime factorizations can be used to obtain alternative characterizations of internal stability conditions. Consider again the standard stability analysis diagram in Figure 5.2. We begin with any rcf’s and lcf’s of P and K̂: P = N M −1 = M̃ −1 Ñ (5.10) K̂ = U V −1 = Ṽ −1 Ũ . (5.11) Lemma 5.7 Consider the system in Figure 5.2. The following conditions are equivalent: 1. The feedback system is internally stable.   M U 2. is invertible in RH∞ . N V   Ṽ −Ũ is invertible in RH∞ . 3. −Ñ M̃ 4. M̃V − ÑU is invertible in RH∞ . 5. Ṽ M − ŨN is invertible in RH∞ . Proof. Note that the system is internally stable if  I −P −K̂ I  K̂ I or, equivalently, Now  I P K̂ I so that Since the matrices  =   I P I P I N M −1 K̂ I −1  M 0 −1 U V −1 I =  0 V −1 M 0  ∈ RH∞ ∈ RH∞ = 0 V  M N    M , N (5.12) U V M N U V  U V M −1 0 0 V −1  −1  are right coprime (this fact is left as an exercise for the reader), equation (5.12) holds iff  −1 M U ∈ RH∞ N V 5.4. Coprime Factorization over RH∞ 75 This proves the equivalence of conditions 1 and 2. The equivalence of conditions 1 and 3 is proved similarly. Conditions 4 and 5 are implied by conditions 2 and 3 from the following equation:  Ṽ −Ñ −Ũ M̃  M N U V  =  Ṽ M − Ũ N 0 0 M̃ V − Ñ U  Since the left-hand side of the above equation is invertible in RH∞ , so is the right-hand side. Hence, conditions 4 and 5 are satisfied. We only need to show that either condition 4 or condition 5 implies condition 1. Let us show that condition 5 implies condition 1; this is obvious since  I P K̂ I −1 = = if  Ṽ M N Ũ I −1   Ṽ −1 Ũ I  0 Ṽ M I N I N M −1 M 0 −1 Ũ I −1  Ṽ 0 0 I  ∈ RH∞ ∈ RH∞ or if condition 5 is satisfied. ✷ Combining Lemma 5.7 and Theorem 5.6, we have the following corollary. Corollary 5.8 Let P be a proper real rational matrix and P = N M −1 = M̃ −1 Ñ be the corresponding rcf and lcf over RH∞ . Then there exists a controller K̂0 = U0 V0−1 = Ṽ0−1 Ũ0 with U0 , V0 , Ũ0 , and Ṽ0 in RH∞ such that  Ṽ0 −Ñ −Ũ0 M̃  M N U0 V0  =  I 0 0 I  (5.13) Furthermore, let F and L be such that A+BF and A+LC are stable. Then a particular set of state-space realizations for these matrices can be given by   Ṽ0 −Ñ M N U0 V0 −Ũ0 M̃    A + BF = F C + DF  A + LC = F C B I D  −L 0  I  −(B + LD) L I 0  −D I (5.14) (5.15) 76 INTERNAL STABILITY Proof. The idea behind the choice of these matrices is as follows. Using the observer theory, find a controller K̂0 achieving internal stability; for example   A + BF + LC + LDF −L (5.16) K̂0 := F 0 Perform factorizations K̂0 = U0 V0−1 = Ṽ0−1 Ũ0 , which are analogous to the ones performed on P . Then Lemma 5.7 implies that each of the two left-hand side block matrices of equation (5.13) must be invertible in RH∞ . In fact, equation (5.13) is satisfied by comparing it with equation (5.7). ✷ Finding a coprime factorization for a scalar transfer function is fairly easy. Let P (s) = num(s)/den(s) where num(s) and den(s) are the numerator and the denominator polynomials of P (s), and let α(s) be a stable polynomial of the same order as den(s). Then P (s) = n(s)/m(s) with n(s) = num(s)/α(s) and m(s) = den(s)/α(s) is a coprime factorization. However, finding an x(s) ∈ H∞ and a y(s) ∈ H∞ such that x(s)n(s) + y(s)m(s) = 1 needs much more work. s−2 and α = (s + 1)(s + 3). Then P (s) = n(s)/m(s) s(s + 3) s−2 s with n(s) = and m(s) = forms a coprime factorization. To find an (s + 1)(s + 3) s+1 x(s) ∈ H∞ and a y(s) ∈ H∞ such that x(s)n(s) + y(s)m(s) = 1, consider a stabilizing s−1 . Then K̂ = u/v with u = K̂ and v = 1 is a coprime controller for P : K̂ = − s + 10 factorization and Example 5.2 Let P (s) = m(s)v(s) − n(s)u(s) = (s + 11.7085)(s + 2.214)(s + 0.077) =: β(s) (s + 1)(s + 3)(s + 10) Then we can take x(s) = −u(s)/β(s) = y(s) = v(s)/β(s) = (s − 1)(s + 1)(s + 3) (s + 11.7085)(s + 2.214)(s + 0.077) (s + 1)(s + 3)(s + 10) (s + 11.7085)(s + 2.214)(s + 0.077) Matlab programs can be used to find the appropriate F and L matrices in statespace so that the desired coprime factorization can be obtained. Let A ∈ Rn×n , B ∈ Rn×m and C ∈ Rp×n . Then an F and an L can be obtained from ≫ F=-lqr(A, B, eye(n), eye(m)); % or 5.5. Notes and References 77 ≫ F=-place(A, B, Pf ); % Pf= poles of A+BF ≫ L = −lqr(A′ , C′ , eye(n), eye(p))′ ; % or ≫ L = −place(A′ , C′ , Pl)′ ; % Pl=poles of A+LC. 5.5 Notes and References The presentation of this chapter is based primarily on Doyle [1984]. The discussion of internal stability and coprime factorization can also be found in Francis [1987], Vidyasagar [1985], and Nett, Jacobson, and Balas [1984]. 5.6 Problems Problem 5.1 Recall that a feedback system is said to be internally stable if all closedloop transfer functions are stable. Describe the conditions for internal stability of the following feedback system: r ✲ d ✲ G1 −✻ d ❄ ✲ d ✲ G2 H ✛ n ✛ d❄ How can the stability conditions be simplified if H(s) and G1 (s) are both stable?  −1 I −K̂ Problem 5.2 Show that ∈ RH∞ if and only if −P I       A B Ĉ B D̂ Ā := (I − DD̂)−1 C DĈ + B̂ 0  is stable. Problem 5.3 Suppose N, M, U, V ∈ RH∞ and N M −1 and U V −1 are right coprime factorizations, respectively. Show that   −1 M 0 M U 0 V N V is also a right coprime factorization. s−1 . Find a stable coprime factorization G = (s + 2)(s − 3) n(s)/m(s) and x, y ∈ RH∞ such that xn + ym = 1. Problem 5.4 Let G(s) = 78 INTERNAL STABILITY (s − 3)(s + α) (s − 1)(s + α) and M (s) = . Show that (s + 2)(s + 3)(s + β) (s + 3)(s + β) (N, M ) is also a coprime factorization of the G in Problem 5.4 for any α > 0 and β > 0. Problem 5.5 Let N (s) = Problem 5.6 Let G = N M −1 be a right coprime factorization over RH∞ . It is called a normalized coprime factorization if N ∼ N + M ∼ M = I. Now consider scalar transfer function G. Then the following procedure can be used to find a normalized coprime factorization: (a) Let G = n/m be any coprime factorization over RH∞ . (b) Find a stable and minimum phase spectral factor w such that w∼ w = n∼ n + m∼ m. Let N = n/w and M = m/w; then G = N/M is a normalized coprime factorization. Find a normalized coprime factorization for Problem 5.4. Problem 5.7 The following procedure constructs a normalized right coprime factorization when G is strictly proper: 1. Get a stabilizable, detectable realization A, B, C. 2. Do the Matlab command F = −lqr(A, B, C ′ C, I). 3. Set  N M   A + BF (s) =  C F  B 0  I Verify that the procedure produces factors that satisfy G = N M −1 . Now try the procedure on   1 1  s−1 s−2   G(s) =   2 1  s s+2 Verify numerically that N (jω)∗ N (jω) + M (jω)∗ M (jω) = I, ∀ω. (5.17) Problem 5.8 Use the procedure in Problem 5.7 to find the normalized right coprime factorization for   1 s+3  s + 1 (s + 1)(s − 2)   G1 (s) =    10 5 G2 (s) =  s−2 2(s + 1)(s + 2) s(s + 3)(s + 4) s+3 s+2 (s + 1)(s + 3)  5.6. Problems 79  −1 −2 1 1 2 3  0 2 −1 3 2 1  −4 −3 −2 1 1 1 G3 (s) =    1 1 1 0 0 0 2 3 4 0 0 0   −1 −2 1 2  0 1 2 1   G4 (s) =   1 1 0 0  1 1 0 0       Problem 5.9 Define the normalized left coprime factorization and describe a procedure to find such factorizations for strictly proper transfer matrices. 80 INTERNAL STABILITY Chapter 6 Performance Specifications and Limitations In this chapter, we consider further the feedback system properties and discuss how to achieve desired performance using feedback control. We also consider the mathematical formulations of optimal H2 and H∞ control problems. A key step in the optimal control design is the selection of weighting functions. We shall give some guidelines to such selection process using some SISO examples. We shall also discuss in some detail the design limitations imposed by bandwidth constraints, the open-loop right-half plane zeros, and the open-loop right-half plane poles using Bode’s gain and phase relation, Bode’s sensitivity integral relation, and the Poisson integral formula. 6.1 Feedback Properties In this section, we discuss the properties of a feedback system. In particular, we consider the benefit of the feedback structure and the concept of design tradeoffs for conflicting objectives — namely, how to achieve the benefits of feedback in the face of uncertainties. r✲e − ✻ ✲ K u di u ❄ ✲ e p✲ P d ❄ ✲ e y✲ n ❄ e✛ Figure 6.1: Standard feedback configuration 81 82 PERFORMANCE SPECIFICATIONS AND LIMITATIONS Consider again the feedback system shown in Figure 5.1. For convenience, the system diagram is shown again in Figure 6.1. For further discussion, it is convenient to define the input loop transfer matrix, Li , and output loop transfer matrix, Lo , as Li = KP, Lo = P K, respectively, where Li is obtained from breaking the loop at the input (u) of the plant while Lo is obtained from breaking the loop at the output (y) of the plant. The input sensitivity matrix is defined as the transfer matrix from di to up : Si = (I + Li )−1 , up = Si di . The output sensitivity matrix is defined as the transfer matrix from d to y: So = (I + Lo )−1 , y = So d. The input and output complementary sensitivity matrices are defined as Ti = I − Si = Li (I + Li )−1 To = I − So = Lo (I + Lo )−1 , respectively. (The word complementary is used to signify the fact that T is the complement of S, T = I − S.) The matrix I + Li is called the input return difference matrix and I + Lo is called the output return difference matrix. It is easy to see that the closed-loop system, if it is internally stable, satisfies the following equations: y = To (r − n) + So P di + So d r − y = So (r − d) + To n − So P di u = KSo (r − n) − KSo d − Ti di up = KSo (r − n) − KSo d + Si di . (6.1) (6.2) (6.3) (6.4) These four equations show the fundamental benefits and design objectives inherent in feedback loops. For example, equation (6.1) shows that the effects of disturbance d on the plant output can be made “small” by making the output sensitivity function So small. Similarly, equation (6.4) shows that the effects of disturbance di on the plant input can be made small by making the input sensitivity function Si small. The notion of smallness for a transfer matrix in a certain range of frequencies can be made explicit using frequency-dependent singular values, for example, σ(So ) < 1 over a frequency range would mean that the effects of disturbance d at the plant output are effectively desensitized over that frequency range. Hence, good disturbance rejection at the plant output (y) would require that
1 σ(So ) = σ (I + P K)−1 = (for disturbance at plant output, d), σ(I + P K)
σ(So P ) = σ (I + P K)−1 P = σ(P Si ) (for disturbance at plant input, di ) 6.1. Feedback Properties 83 be made small and good disturbance rejection at the plant input (up ) would require that
1 σ(Si ) = σ (I + KP )−1 = (for disturbance at plant input, di ), σ(I + KP )
σ(Si K) = σ K(I + P K)−1 = σ(KSo ) (for disturbance at plant output, d) be made small, particularly in the low-frequency range where d and di are usually significant. Note that σ(P K) − 1 σ(KP ) − 1 ≤ σ(I + P K) ≤ σ(P K) + 1 ≤ σ(I + KP ) ≤ σ(KP ) + 1 then 1 1 ≤ σ(So ) ≤ , if σ(P K) > 1 σ(P K) + 1 σ(P K) − 1 1 1 ≤ σ(Si ) ≤ , if σ(KP ) > 1 σ(KP ) + 1 σ(KP ) − 1 These equations imply that σ(So ) ≪ 1 ⇐⇒ σ(P K) ≫ 1 σ(Si ) ≪ 1 ⇐⇒ σ(KP ) ≫ 1. Now suppose P and K are invertible; then σ(P K) ≫ 1 or σ(KP ) ≫ 1 ⇐⇒ σ(P K) ≫ 1 or σ(KP ) ≫ 1 ⇐⇒ 1 σ(K)
1 σ(KSo ) = σ K(I + P K)−1 ≈ σ(P −1 ) = σ(P )
σ(So P ) = σ (I + P K)−1 P ≈ σ(K −1 ) = Hence good performance at plant output (y) requires, in general, large output loop gain σ(Lo ) = σ(P K) ≫ 1 in the frequency range where d is significant for desensitizing d and large enough controller gain σ(K) ≫ 1 in the frequency range where di is significant for desensitizing di . Similarly, good performance at plant input (up ) requires, in general, large input loop gain σ(Li ) = σ(KP ) ≫ 1 in the frequency range where di is significant for desensitizing di and large enough plant gain σ(P ) ≫ 1 in the frequency range where d is significant, which cannot be changed by controller design, for desensitizing d. [In general, So 6= Si unless K and P are square and diagonal, which is true if P is a scalar system. Hence, small σ(So ) does not necessarily imply small σ(Si ); in other words, good disturbance rejection at the output does not necessarily mean good disturbance rejection at the plant input.] Hence, good multivariable feedback loop design boils down to achieving high loop (and possibly controller) gains in the necessary frequency range. 84 PERFORMANCE SPECIFICATIONS AND LIMITATIONS Despite the simplicity of this statement, feedback design is by no means trivial. This is true because loop gains cannot be made arbitrarily high over arbitrarily large frequency ranges. Rather, they must satisfy certain performance tradeoff and design limitations. A major performance tradeoff, for example, concerns commands and disturbance error reduction versus stability under the model uncertainty. Assume that the plant model is perturbed to (I + ∆)P with ∆ stable, and assume that the system is nominally stable (i.e., the closed-loop system with ∆ = 0 is stable). Now the perturbed closed-loop system is stable if det (I + (I + ∆)P K) = det(I + P K) det(I + ∆To ) has no right-half plane zero. This would, in general, amount to requiring that k∆To k be small or that σ̄(To ) be small at those frequencies where ∆ is significant, typically at high-frequency range, which, in turn, implies that the loop gain, σ(Lo ), should be small at those frequencies. Still another tradeoff is with the sensor noise error reduction. The conflict between the disturbance rejection and the sensor noise reduction is evident in equation (6.1). Large σ(Lo (jω)) values over a large frequency range make errors due to d small. However, they also make errors due to n large because this noise is “passed through” over the same frequency range, that is, y = To (r − n) + So P di + So d ≈ (r − n) Note that n is typically significant in the high-frequency range. Worst still, large loop gains outside of the bandwidth of P — that is, σ(Lo (jω)) ≫ 1 or σ(Li (jω)) ≫ 1 while σ(P (jω)) ≪ 1 — can make the control activity (u) quite unacceptable, which may cause the saturation of actuators. This follows from u = KSo (r − n − d) − Ti di = Si K(r − n − d) − Ti di ≈ P −1 (r − n − d) − di Here, we have assumed P to be square and invertible for convenience. The resulting equation shows that disturbances and sensor noise are actually amplified at u whenever the frequency range significantly exceeds the bandwidth of P , since for ω such that σ(P (jω)) ≪ 1 we have 1 σ[P −1 (jω)] = ≫1 σ[P (jω)] Similarly, the controller gain, σ(K), should also be kept not too large in the frequency range where the loop gain is small in order not to saturate the actuators. This is because for small loop gain σ(Lo (jω)) ≪ 1 or σ(Li (jω)) ≪ 1 u = KSo (r − n − d) − Ti di ≈ K(r − n − d) Therefore, it is desirable to keep σ(K) not too large when the loop gain is small. To summarize the above discussion, we note that good performance requires in some frequency range, typically some low-frequency range (0, ωl ), σ(P K) ≫ 1, σ(KP ) ≫ 1, σ(K) ≫ 1 6.2. Weighted H2 and H∞ Performance 85 and good robustness and good sensor noise rejection require in some frequency range, typically some high-frequency range (ωh , ∞), σ(P K) ≪ 1, σ(KP ) ≪ 1, σ(K) ≤ M where M is not too large. These design requirements are shown graphically in Figure 6.2. The specific frequencies ωl and ωh depend on the specific applications and the knowledge one has of the disturbance characteristics, the modeling uncertainties, and the sensor noise levels. ❍❍ ✻ ❅ ❍❍ ❍❍ ❅ ❅ ❅ ❅ σ(L) ❳❳ ❳❳ ❅ ❅ ❩ ❅ ❅ ❩ ❅ ❅ ❩ ❅ ❩ ❙ ❅ ❩ ❅ ❙ ❩ ❩ ❙ ωh ❩ ❍ ✲ ❩ ❍❍ ωl log ω ❍❍ ❩❩ ✓ ❅❍❍ ❙ σ(L) ❙ ❅ ❍❍ ❍ ❅ ❙ ❍❍ ❅ ❙ ❍ ❍ ❅ ❙ ❅ ❙ Figure 6.2: Desired loop gain 6.2 Weighted H2 and H∞ Performance In this section, we consider how to formulate some performance objectives into mathematically tractable problems. As shown in Section 6.1, the performance objectives of a feedback system can usually be specified in terms of requirements on the sensitivity functions and/or complementary sensitivity functions or in terms of some other closedloop transfer functions. For instance, the performance criteria for a scalar system may be specified as requiring  |S(jω)| ≤ ε, ∀ω ≤ ω0 , |S(jω)| ≤ M, ∀ω > ω0 where S(jω) = 1/(1 + P (jω)K(jω)). However, it is much more convenient to reflect the system performance objectives by choosing appropriate weighting functions. For 86 PERFORMANCE SPECIFICATIONS AND LIMITATIONS example, the preceding performance objective can be written as |We (jω)S(jω)| ≤ 1, ∀ω with |We (jω)| =  1/ε, ∀ω ≤ ω0 1/M, ∀ω > ω0 To use We in control design, a rational transfer function We (s) is usually used to approximate the foregoing frequency response. The advantage of using weighted performance specifications is obvious in multivariable system design. First, some components of a vector signal are usually more important than others. Second, each component of the signal may not be measured in the same units; for example, some components of the output error signal may be measured in terms of length, and others may be measured in terms of voltage. Therefore, weighting functions are essential to make these components comparable. Also, we might be primarily interested in rejecting errors in a certain frequency range (for example, low frequencies); hence some frequency-dependent weights must be chosen. d˜i ❄ Wi ũ ✻ Wu ✻ ✲ Wr r✲e − ✻ ✲ K u di ✲❄ e d˜ ❄ Wd ✲ P d ✲❄ e y✲ We e✲ n ❄ ñ e✛ Wn ✛ Figure 6.3: Standard feedback configuration with weights In general, we shall modify the standard feedback diagram in Figure 6.1 into Figure 6.3. The weighting functions in Figure 6.3 are chosen to reflect the design objectives and knowledge of the disturbances and sensor noise. For example, Wd and Wi may be chosen to reflect the frequency contents of the disturbances d and di or they may be used to model the disturbance power spectrum depending on the nature of signals involved in the practical systems. The weighting matrix Wn is used to model the frequency contents of the sensor noise while We may be used to reflect the requirements on the shape of certain closed-loop transfer functions (for example, the shape of the output sensitivity function). Similarly, Wu may be used to reflect some restrictions on the control or actuator signals, and the dashed precompensator Wr is an optional element used to achieve deliberate command shaping or to represent a nonunity feedback system in equivalent unity feedback form. 6.2. Weighted H2 and H∞ Performance 87 It is, in fact, essential that some appropriate weighting matrices be used in order to utilize the optimal control theory discussed in this book (i.e., H2 and H∞ theory). So a very important step in the controller design process is to choose the appropriate weights, We , Wd , Wu , and possibly Wn , Wi , Wr . The appropriate choice of weights for a particular practical problem is not trivial. In many occasions, as in the scalar case, the weights are chosen purely as a design parameter without any physical bases, so these weights may be treated as tuning parameters that are chosen by the designer to achieve the best compromise between the conflicting objectives. The selection of the weighting matrices should be guided by the expected system inputs and the relative importance of the outputs. Hence, control design may be regarded as a process of choosing a controller K such that certain weighted signals are made small in some sense. There are many different ways to define the smallness of a signal or transfer matrix, as we have discussed in the last chapter. Different definitions lead to different control synthesis methods, and some are much harder than others. A control engineer should make a judgment of the mathematical complexity versus engineering requirements. Next, we introduce two classes of performance formulations: H2 and H∞ criteria. For the simplicity of presentation, we shall assume that di = 0 and n = 0. H2 Performance Assume, for example, that the disturbance d˜ can be approximately modeled as an impulse with random input direction; that is, ˜ = ηδ(t) d(t) and E(ηη ∗ ) = I where E denotes the expectation. We may choose to minimize the expected energy of ˜ the error e due to the disturbance d: Z ∞  n o 2 2 2 E kek2 = E kek dt = kWe So Wd k2 0 In general, a controller minimizing only the above criterion can lead to a very large control signal u that could cause saturation of the actuators as well as many other undesirable problems. Hence, for a realistic controller design, it is necessary to include the control signal u in the cost function. Thus, our design criterion would usually be something like this:  2  n o We So Wd 2 2 E kek2 + ρ2 kũk2 = ρWu KSo Wd 2 with some appropriate choice of weighting matrix Wu and scalar ρ. The parameter ρ clearly defines the tradeoff we discussed earlier between good disturbance rejection at 88 PERFORMANCE SPECIFICATIONS AND LIMITATIONS the output and control effort (or disturbance and sensor noise rejection at the actuators). Note that ρ can be set to ρ = 1 by an appropriate choice of Wu . This problem can be viewed as minimizing the energy consumed by the system in order to reject the disturbance d. This type of problem was the dominant paradigm in the 1960s and 1970s and is usually referred to as linear quadratic Gaussian control, or simply as LQG. (Such problems will also be referred to as H2 mixed-sensitivity problems for consistency with the H∞ problems discussed next.) The development of this paradigm stimulated extensive research efforts and is responsible for important technological innovation, particularly in the area of estimation. The theoretical contributions include a deeper understanding of linear systems and improved computational methods for complex systems through state-space techniques. The major limitation of this theory is the lack of formal treatment of uncertainty in the plant itself. By allowing only additive noise for uncertainty, the stochastic theory ignored this important practical issue. Plant uncertainty is particularly critical in feedback systems. (See Paganini [1995,1996] for some recent results on robust H2 control theory.) H∞ Performance Although the H2 norm (or L2 norm) may be a meaningful performance measure and although LQG theory can give efficient design compromises under certain disturbance and plant assumptions, the H2 norm suffers a major deficiency. This deficiency is due to the fact that the tradeoff between disturbance error reduction and sensor noise error reduction is not the only constraint on feedback design. The problem is that these performance tradeoffs are often overshadowed by a second limitation on high loop gains — namely, the requirement for tolerance to uncertainties. Though a controller may be designed using FDLTI models, the design must be implemented and operated with a real physical plant. The properties of physical systems (in particular, the ways in which they deviate from finite-dimensional linear models) put strict limitations on the frequency range over which the loop gains may be large. A solution to this problem would be to put explicit constraints on the loop gain in the cost function. For instance, one may chose to minimize sup kek2 = kWe So Wd k∞ kd˜k2 ≤1 subject to some restrictions on the control energy or control bandwidth: sup kũk2 = kWu KSo Wd k∞ kd̃k2 ≤1 Or, more frequently, one may introduce a parameter ρ and a mixed criterion  2  o n We So Wd sup kek22 + ρ2 kũk22 = ρWu KSo Wd ∞ kd̃k2 ≤1 6.3. Selection of Weighting Functions 89 Alternatively, if the system robust stability margin is the major concern, the weighted complementary sensitivity has to be limited. Thus the whole cost function may be   We So Wd ρW1 To W2 ∞ where W1 and W2 are the frequency-dependent uncertainty scaling matrices. These design problems are usually called H∞ mixed-sensitivity problems. For a scalar system, an H∞ norm minimization problem can also be viewed as minimizing the maximum magnitude of the system’s steady-state response with respect to the worst-case sinusoidal inputs. 6.3 Selection of Weighting Functions The selection of weighting functions for a specific design problem often involves ad hoc fixing, many iterations, and fine tuning. It is very hard to give a general formula for the weighting functions that will work in every case. Nevertheless, we shall try to give some guidelines in this section by looking at a typical SISO problem. Consider an SISO feedback system shown in Figure 6.1. Then the tracking error is e = r − y = S(r − d) + T n − SP di . So, as we have discussed earlier, we must keep |S| small over a range of frequencies, typically low frequencies where r and d are significant. To motivate the choice of our performance weighting function We , let L = P K be a standard second-order system ωn2 L= s(s + 2ξωn ) It is well-known from the classical control theory that the quality of the (step) time response can be quantified by rise time tr , settling time ts , and percent overshoot 100Mp%. Furthermore, these performance indices can be approximately calculated as tr ≈ − √ πξ 4 0.6 + 2.16ξ , 0.3 ≤ ξ ≤ 0.8; ts ≈ ; Mp = e 1−ξ2 , 0 < ξ < 1 ωn ξωn The key points to note are that (1) the speed of the system response is proportional to ωn and (2) the overshoot of the system response is determined only by the damping ratio ξ. It is well known that the frequency ωn and the damping ratio ξ can be essentially captured in the frequency domain by the open-loop crossover frequency and the phase margin or the bandwidth and the resonant peak of the closed-loop complementary sensitivity function T . Since our performance objectives are closely related to the sensitivity function, we shall consider in some detail how these time domain indices or, equivalently, ωn and ξ are related to the frequency response of the sensitivity function S= s(s + 2ξωn ) 1 = 2 1+L s + 2ξωn s + ωn2 90 PERFORMANCE SPECIFICATIONS AND LIMITATIONS 1 10 0 sensitivity function 10 −1 10 −2 10 −1 10 0 10 normalized frequency 1 10 Figure 6.4: Sensitivity function S for ξ = 0.05, 0.1, 0.2, 0.5, 0.8, and 1 with normalized frequency (ω/ωn ) The frequency response of the sensitivity function S is shown in Figure √ 6.4. Note that √ |S(jωn / 2)| = 1. We can regard the closed-loop bandwidth ωb ≈ ωn / 2, since beyond this frequency the closed-loop system will not be able to track the reference and the disturbance will actually be amplified. Next, note that p α α2 + 4ξ 2 Ms := kSk∞ = |S(jωmax )| = p (1 − α2 )2 + 4ξ 2 α2 q p where α = 0.5 + 0.5 1 + 8ξ 2 and ωmax = αωn . For example, Ms = 5.123 when ξ = 0.1. The relationship between ξ and Ms is shown in Figure 6.5. It is clear that the overshoot can be excessive if Ms is large. Hence a good control design should not have a very large Ms . Now suppose we are given the time domain performance specifications then we can determine the corresponding requirements in frequency domain in terms of the bandwidth ωb and the peak sensitivity Ms . Hence a good control design should result in a sensitivity function S satisfying both the bandwidth ωb and the peak sensitivity Ms requirements, as shown in Figure 6.6. These requirements can be approximately represented as s , s = jω, ∀ ω |S(s)| ≤ s/Ms + ωb 6.3. Selection of Weighting Functions 91 5 4.5 peak sensitivity 4 3.5 3 2.5 2 1.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 damping ratio 0.7 0.8 0.9 Figure 6.5: Peak sensitivity Ms versus damping ratio ξ 1/|We| Ms ωb |S(jω)| 1 Figure 6.6: Performance weight We and desired S 1 92 PERFORMANCE SPECIFICATIONS AND LIMITATIONS Or, equivalently, |We S| ≤ 1 with We = s/Ms + ωb s (6.5) The preceding discussion applies in principle to most control design and hence the preceding weighting function can, in principle, be used as a candidate weighting function in an initial design. Since the steady-state error with respect to a step input is given by |S(0)|, it is clear that |S(0)| = 0 if the closed-loop system is stable and kWe Sk∞ < ∞. Unfortunately, the optimal control techniques described in this book cannot be used directly for problems with such weighting functions since these techniques assume that all unstable poles of the system (including plant and all performance and control weighting functions) are stabilizable by the control and detectable from the measurement outputs, which is clearly not satisfied if We has an imaginary axis pole since We is not detectable from the measurement. We shall discuss in Chapter 14 how such problems can be reformulated so that the techniques described in this book can be applied. A theory dealing directly with such problems is available but is much more complicated both theoretically and computationally and does not seem to offer much advantage. 1/|We| Ms ωb |S(j ω)| 1 ε Figure 6.7: Practical performance weight We and desired S Now instead of perfect tracking for step input, suppose we only need the steadystate error with respect to a step input to be no greater than ǫ (i.e., |S(0)| ≤ ǫ); then it is sufficient to choose a weighting function We satisfying |We (0)| ≥ 1/ε so that kWe Sk∞ ≤ 1 can be achieved. A possible choice of We can be obtained by modifying the weighting function in equation (6.5): We = s/Ms + ωb s + ωb ε (6.6) 6.3. Selection of Weighting Functions 93 Hence, for practical purpose, one can usually choose a suitable ε, as shown in Figure 6.7, to satisfy the performance specifications. If a steeper transition between low-frequency and high-frequency is desired, the weight We can be modified as follows: We =  √ k s/ k Ms + ωb √ s + ωb k ε (6.7) for some integer k ≥ 1. The selection of control weighting function Wu follows similarly from the preceding discussion by considering the control signal equation u = KS(r − n − d) − T di The magnitude of |KS| in the low-frequency range is essentially limited by the allowable cost of control effort and saturation limit of the actuators; hence, in general, the maximum gain Mu of KS can be fairly large, while the high-frequency gain is essentially limited by the controller bandwidth (ωbc ) and the (sensor) noise frequencies. Ideally, one would like to roll off as fast as possible beyond the desired control bandwidth so that the high-frequency noises are attenuated as much as possible. Hence a candidate weight Wu would be s + ωbc /Mu (6.8) Wu = ωbc 1/|Wu | Mu |KS(jω)| 1 ω bc ε1 Figure 6.8: Control weight Wu and desired KS However, again the optimal control design techniques developed in this book cannot be applied directly to a problem with an improper control weighting function. Hence 94 PERFORMANCE SPECIFICATIONS AND LIMITATIONS we shall introduce a far away pole to make Wu proper: Wu = s + ωbc /Mu ε1 s + ωbc (6.9) for a small ε1 > 0, as shown in Figure 6.8. Similarly, if a faster rolloff is desired, one may choose √  k s + ωbc / k Mu Wu = (6.10) √ k ε s + ω 1 bc for some integer k ≥ 1. The weights for MIMO problems can be initially chosen as diagonal matrices with each diagonal term chosen in the foregoing form. 6.4 Bode’s Gain and Phase Relation One important problem that arises frequently is concerned with the level of performance that can be achieved in feedback design. It has been shown in Section 6.1 that the feedback design goals are inherently conflicting, and a tradeoff must be performed among different design objectives. It is also known that the fundamental requirements, such as stability and robustness, impose inherent limitations on the feedback properties irrespective of design methods, and the design limitations become more severe in the presence of right-half plane zeros and poles in the open-loop transfer function. In the classical feedback theory, Bode’s gain-phase integral relation (see Bode [1945]) has been used as an important tool to express design constraints in scalar systems. This integral relation says that the phase of a stable and minimum phase transfer function is determined uniquely by the magnitude of the transfer function. More precisely, let L(s) be a stable and minimum phase transfer function: then ∠L(jω0 ) = 1 π Z ∞ −∞ |ν| d ln |L| ln coth dν dν 2 where ν := ln(ω/ω0 ). The function ln coth ure 6.9. (6.11) e|ν|/2 + e−|ν|/2 |ν| = ln |ν|/2 is plotted in Fig2 e − e−|ν|/2 |ν| decreases rapidly as ω deviates from ω0 and hence the integral 2 d ln |L(jω)| near the frequency ω0 . This is clear depends mostly on the behavior of dν from the following integration:   Z  1.1406 (rad), α = ln 3  65.3o , α = ln 3 1 α |ν| 1.3146 (rad), α = ln 5 = 75.3o , α = ln 5 dν = ln coth   π −α 2 1.443 (rad), α = ln 10 82.7o , α = ln 10. Note that ln coth 6.4. Bode’s Gain and Phase Relation 95 5 4.5 4 3.5 ln coth |ν |/ 2 3 2.5 2 1.5 1 0.5 0 −3 −2 −1 ν 0 1 2 Figure 6.9: The function ln coth 3 |ν| vs ν 2 d ln |L(jω)| is the slope of the Bode plot, which is generally negative for dν almost all frequencies. It follows that ∠L(jω0 ) will be large if the gain L attenuates slowly near ω0 and small if it attenuates rapidly near ω0 . For example, suppose the d ln |L(jω)| = −ℓ; that is, (−20ℓ dB per decade), in the neighborhood of ω0 ; then slope dν it is reasonable to expect  ω 1 o  −ℓ × 65.3 , if the slope of L = −ℓ for 3 ≤ ω0 ≤ 3 1 o −ℓ × 75.3 , if the slope of L = −ℓ for 5 ≤ ωω0 ≤ 5 ∠L(jω0 ) <  1 ≤ ωω0 ≤ 10. −ℓ × 82.7o , if the slope of L = −ℓ for 10 Note that The behavior of ∠L(jω) is particularly important near the crossover frequency ωc , where |L(jωc )| = 1 since π + ∠L(jωc ) is the phase margin of the feedback system. Further, the return difference is given by |1 + L(jωc )| = |1 + L−1 (jωc )| = 2 sin π + ∠L(jωc ) , 2 which must not be too small for good stability robustness. If π + ∠L(jωc ) is forced to be very small by rapid gain attenuation, the feedback system will amplify disturbances and exhibit little uncertainty tolerance at and near ωc . Since it is generally required that the loop transfer function L roll off as fast as possible in the high-frequency range, it is reasonable to expect that ∠L(jωc ) is at most −ℓ × 90o if the slope of L(jω) is −ℓ near ωc . Thus it is important to keep the slope of L near ωc not much smaller than −1 for a reasonably wide range of frequencies in order to guarantee some reasonable 96 PERFORMANCE SPECIFICATIONS AND LIMITATIONS performance. The conflict between attenuation rate and loop quality near crossover is thus clearly evident. Bode’s gain and phase relation can be extended to stable and nonminimum phase transfer functions easily. Let z1 , z2 , . . . , zk be the right-half plane zeros of L(s), then L can be factorized as −s + zk −s + z1 −s + z2 ··· Lmp (s) L(s) = s + z1 s + z2 s + zk where Lmp is stable and minimum phase and |L(jω)| = |Lmp (jω)|. Hence ∠L(jω0 ) = ∠Lmp (jω0 ) + ∠ k Y −jω0 + zi i=1 = 1 π Z ∞ −∞ which gives jω0 + zi k X −jω0 + zi |ν| d ln |Lmp | ∠ , ln coth dν + dν 2 jω0 + zi i=1 1 ∠L(jω0 ) = π Z ∞ −∞ k X −jω0 + zi d ln |L| |ν| ∠ . ln coth dν + dν 2 jω0 + zi i=1 (6.12) −jω0 + zi ≤ 0 for each i, a nonminimum phase zero contributes an additional jω0 + zi phase lag and imposes limitations on the rolloff rate of the open-loop gain. For example, suppose L has a zero at z > 0; then Since ∠ φ1 (ω0 /z) := ∠ −jω0 + z jω0 + z ω0 =z,z/2,z/4 = −90o , −53.13o, −28o, as shown in Figure 6.10. Since the slope of |L| near the crossover frequency is, in general, no greater than −1, which means that the phase due to the minimum phase part, Lmp , of L will, in general, be no greater than −90o , the crossover frequency (or the closed-loop bandwidth) must satisfy ωc < z/2 (6.13) in order to guarantee the closed-loop stability and some reasonable closed-loop performance. Next suppose L has a pair of complex right-half zeros at z = x ± jy with x > 0; then φ2 (ω0 /|z|) := ∠ −jω0 + z −jω0 + z̄ jω0 + z jω0 + z̄ ω0 =|z|,|z|/2,|z|/3,|z|/4  −56o , Re(z) ≫ ℑ(z)  −180o , −106.26o, −73.7o , o o o −180 , −86.7 , −55.9 , −41.3o, Re(z) ≈ ℑ(z) ≈  o −360 , 0o , 0o , 0o , Re(z) ≪ ℑ(z) 6.4. Bode’s Gain and Phase Relation 97 0 −10 phase φ1(ω0 /z) (in degree) −20 −30 −40 −50 −60 −70 −80 −90 0 0.1 0.2 0.3 ω0 / |z| 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6.10: Phase φ1 (ω0 /z) due to a real zero z > 0 0 −20 y/x=10 y/x=100 phase φ2 ( ω0 / |z| ) (in degree) −40 −60 y/x=3 y/x=1 −80 −100 −120 −140 y/x=0.01 −160 −180 −200 0 0.1 0.2 0.3 ω0 / |z| 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6.11: Phase φ2 (ω0 /|z|) due to a pair of complex zeros: z = x ± jy and x > 0 98 PERFORMANCE SPECIFICATIONS AND LIMITATIONS as shown in Figure 6.11. In this case we satisfy   |z|/4, |z|/3, ωc <  |z|, conclude that the crossover frequency must Re(z) ≫ ℑ(z) Re(z) ≈ ℑ(z) Re(z) ≪ ℑ(z) (6.14) in order to guarantee the closed-loop stability and some reasonable closed-loop performance. 6.5 Bode’s Sensitivity Integral In this section, we consider the design limitations imposed by the bandwidth constraints and the right-half plane poles and zeros using Bode’s sensitivity integral and Poisson integral. Let L be the open-loop transfer function with at least two more poles than zeros and let p1 , p2 , . . . , pm be the open right-half plane poles of L. Then the following Bode’s sensitivity integral holds: Z 0 ∞ ln |S(jω)|dω = π m X Re(pi ) (6.15) i=1 In the case where L is stable, the integral simplifies to Z ∞ ln |S(jω)|dω = 0 (6.16) 0 These integrals show that there will exist a frequency range over which the magnitude of the sensitivity function exceeds one if it is to be kept below one at other frequencies, as illustrated in Figure 6.12. This is the so-called water bed effect. |S(j ω)| + 1 ω − Figure 6.12: Water bed effect of sensitivity function 6.5. Bode’s Sensitivity Integral 99 Suppose that the feedback system is designed such that the level of sensitivity reduction is given by |S(jω)| ≤ ǫ < 1, ∀ω ∈ [0, ωl ] where ǫ > 0 is a given constant. Bandwidth constraints in feedback design typically require that the open-loop transfer function be small above a specified frequency, and that it roll off at a rate of more than one pole-zero excess above that frequency. These constraints are commonly needed to ensure stability robustness despite the presence of modeling uncertainty in the plant model, particularly at high frequencies. One way of quantifying such bandwidth constraints is by requiring the open-loop transfer function to satisfy |L(jω)| ≤ Mh ≤ ǫ̃ < 1, ω 1+β ∀ω ∈ [ωh , ∞) where ωh > ωl , and Mh > 0, β > 0 are some given constants. Note that for ω ≥ ωh , |S(jω)| ≤ 1 1 ≤ h 1 − |L(jω)| 1 − ωM 1+β and − Z ∞ ωh   Mh ln 1 − 1+β dω ω ∞ Z X = i=1 ∞ ωh 1 i  Mh ω 1+β i dω !i ωh Mh = i i(1 + β) − 1 ωh1+β i=1 !i ! ∞ ωh X 1 ωh Mh Mh ≤ =− ln 1 − 1+β β i=1 i ωh1+β β ωh ωh ln(1 − ǫ̃). ≤ − β ∞ X 1 Then π m X Re(pi ) = Z 0 i=1 ∞ ln |S(jω)|dω Z ∞ ln |S(jω)|dω ln |S(jω)|dω + ωh ωl 0  Z ∞  Mh ≤ ωl ln ǫ + (ωh − ωl ) max ln |S(jω)| − ln 1 − 1+β dω ω∈[ωl ,ωh ] ω ωh ωh ≤ ωl ln ǫ + (ωh − ωl ) max ln |S(jω)| − ln(1 − ǫ̃), ω∈[ωl ,ωh ] β = Z ωl ln |S(jω)|dω + Z ωh 100 PERFORMANCE SPECIFICATIONS AND LIMITATIONS which gives l   ω ω−ω ωh 1 h l max |S(jω)| ≥ e (1 − ǫ̃) β(ωh −ωl ) ǫ ω∈[ωl ,ωh ] α where α= π Pm i=1 Re(pi ) . ωh − ωl The above lower bound shows that the sensitivity can be very significant in the transition band. Next, using the Poisson integral relation, we investigate the design constraints on sensitivity properties imposed by open-loop nonminimum phase zeros. Suppose L has at least one more poles than zeros and suppose z = x0 + jy0 with x0 > 0 is a right-half plane zero of L. Then Z ∞ −∞ ln |S(jω)| m Y z + pi x0 dω = π ln x20 + (ω − y0 )2 z − pi i=1 (6.17) This integral implies that the sensitivity reduction ability of the system may be severely limited by the open-loop unstable poles and nonminimum phase zeros, especially when these poles and zeros are close to each other. Define Z ωl x0 θ(z) := 2 + (ω − y )2 dω x 0 −ωl 0 Then π ln Z ∞ m Y x0 z + pi = ln |S(jω)| 2 dω z − pi x0 + (ω − y0 )2 −∞ i=1 ≤ (π − θ(z)) ln kS(jω)k∞ + θ(z) ln(ǫ), which gives kS(s)k∞ θ(z)   π−θ(z) 1 ≥ ǫ m Y z + pi z − pi i=1 π ! π−θ(z) This lower bound on the maximum sensitivity shows that for a nonminimum phase system, its sensitivity must increase significantly beyond one at certain frequencies if the sensitivity reduction is to be achieved at other frequencies. 6.6 Analyticity Constraints Let p1 , p2 , . . . , pm and z1 , z2 , . . . , zk be the open right-half plane poles and zeros of L, respectively. Suppose that the closed-loop system is stable. Then S(pi ) = 0, T (pi ) = 1, i = 1, 2, . . . , m 6.6. Analyticity Constraints 101 and S(zj ) = 1, T (zj ) = 0, j = 1, 2, . . . , k The internal stability of the feedback system is guaranteed by satisfying these analyticity (or interpolation) conditions. On the other hand, these conditions also impose severe limitations on the achievable performance of the feedback system. Suppose S = (I + L)−1 and T = L(I + L)−1 are stable. Then p1 , p2 , . . . , pm are the right-half plane zeros of S and z1 , z2 , . . . , zk are the right-half plane zeros of T . Let Bp (s) = m Y s − pi i=1 s + pi , Bz (s) = k Y s − zj s + zj j=1 Then |Bp (jω)| = 1 and |Bz (jω)| = 1 for all frequencies and, moreover, Bp−1 (s)S(s) ∈ H∞ , Bz−1 (s)T (s) ∈ H∞ . Hence, by the maximum modulus theorem, we have kS(s)k∞ = Bp−1 (s)S(s) ≥ |Bp−1 (z)S(z)| ∞ for any z with Re(z) > 0. Let z be a right-half plane zero of L; then kS(s)k∞ ≥ |Bp−1 (z)| = m Y z + pi z − pi i=1 kT (s)k∞ ≥ |Bz−1 (p)| = k Y p + zj p − zj j=1 Similarly, one can obtain where p is a right-half plane pole of L. The weighted problem can be considered in the same fashion. Let We be a weight such that We S is stable. Then kWe (s)S(s)k∞ ≥ |We (z)| Now suppose We (s) = Then which gives m Y z + pi z − pi i=1 s/Ms + ωb , kWe Sk∞ ≤ 1, and z is a real right-half plane zero. s + ωb ǫ m Y z − pi z/Ms + ωb =: α, ≤ z + ωb ǫ z + pi i=1 z 1 1 ) ≈ z(α − ) (α − 1 − αǫ Ms Ms where α = 1 if L has no right-half plane poles. This shows that the bandwidth of the closed-loop must be much smaller than the right-half plane zero. Similar conclusions can be arrived at for complex right-half plane zeros. ωb ≤ 102 6.7 PERFORMANCE SPECIFICATIONS AND LIMITATIONS Notes and References The loop-shaping design is well-known for SISO systems in the classical control theory. The idea was extended to MIMO systems by Doyle and Stein [1981] using the LQG design technique. The limitations of the loop-shaping design are discussed in detail in Stein and Doyle [1991]. Chapter 16 presents another loop-shaping method using H∞ control theory, which has the potential to overcome the limitations of the LQG/LTR method. Some additional discussions on the choice of weighting functions can be found in Skogestad and Postlethwaite [1996]. The design tradeoffs and limitations for SISO systems are discussed in detail in Bode [1945], Horowitz [1963], and Doyle, Francis, and Tannenbaum [1992]. The monograph by Freudenberg and Looze [1988] contains many multivariable generalizations. The multivariable generalization of Bode’s integral relation can be found in Chen [1995], on which Section 6.5 is based. Some related results can be found in Boyd and Desoer [1985]. Additional related results can be found in a recent book by Seron, Braslavsky, and Goodwin [1997]. 6.8 Problems Problem 6.1 Let P be an open-loop plant. It is desired to design a controller so that the overshoot ≤ 10% and settling time ≤ 10 sec. Estimate the allowable peak sensitivity Ms and the closed-loop bandwidth. 1 be an open-loop transfer function of a unity feedback s(s + 1)2 system. Find the phase margin, overshoot, settling time, and the corresponding Ms . Problem 6.2 Let L1 = Problem 6.3 Repeat Problem 6.2 with L2 = 100(s + 10) . (s + 1)(s + 2)(s + 20) 10(1 − s) . Use classical loop-shaping method to design a firsts(s + 10) order lead or lag controller so that the system has at least 30o phase margin and as large a crossover frequency as possible. Problem 6.4 Let P = Problem 6.5 Use the root locus method to show that a nonminimum phase system cannot be stabilized by a very high-gain controller. 5 . Design a lead or lag controller so that the system (1 − s)(s + 2) o has at least 30 phase margin with loop gain ≥ 2 for any frequency ω ≤ 0.1 and the smallest possible bandwidth (or crossover frequency). Problem 6.6 Let P = Problem 6.7 Use the root locus method to show that an unstable system cannot be stabilized by a very low gain controller. 6.8. Problems 103 Problem 6.8 Consider the unity-feedback loop with proper controller K(s) and strictly proper plant P (s), both assumed square. Assume internal stability. 1. Let w(s) be a scalar weighting function, assumed in RH∞ . Define ǫ = kw(I + P K)−1 k∞ , δ = kK(I + P K)−1 k∞ so ǫ measures, say, disturbance attenuation and δ measures, say, control effort. Derive the following inequality, which shows that ǫ and δ cannot both be small simultaneously in general. For every Re s0 ≥ 0 |w(s0 )| ≤ ǫ + |w(s0 )|σmin [P (s0 )]δ. 2. If we want very good disturbance attenuation at a particular frequency, you might guess that we need high controller gain at that frequency. Fix ω with jω not a pole of P (s), and suppose ǫ := σmax [(I + P K)−1 (jω)] < 1. Derive a lower bound for σmin [K(jω)]. This lower bound should blow up as ǫ → 0. Problem 6.9 Suppose that P is proper and has one right half plane zero at s = z > 0. w Suppose that y = 1+P K , where w is a unit step at time t = 0, and that our performance specification is  α, if 0 ≤ t ≤ T ; |y(t)| ≤ β, if T < t for some α > 1 > β > 0. Show that for a proper, internally stabilizing, LTI controller K to exist that meets the specification, we must have that   α−β ln ≤ zT. α−1 What tradeoffs does this imply? Problem 6.10 Let K be a stabilizing controller for the plant P = s−α (s − β)(s + γ) α > 0, β > 0, γ ≥ 0. Suppose |S(jω)| ≤ δ < 1, ∀ω ∈ [−ω0 , ω0 ] where S(s) = 1 . 1 + PK Find a lower bound for kSk∞ and calculate the lower bound for α = 1, β = 2, γ = 10, δ = 0.2, and ω0 = 1. 104 PERFORMANCE SPECIFICATIONS AND LIMITATIONS Chapter 7 Balanced Model Reduction Simple linear models/controllers are normally preferred over complex ones in control system design for some obvious reasons: They are much easier to do analysis and synthesis with. Furthermore, simple controllers are easier to implement and are more reliable because there are fewer things to go wrong in the hardware or bugs to fix in the software. In the case when the system is infinite dimensional, the model/controller approximation becomes essential. In this chapter we consider the problem of reducing the order of a linear multivariable dynamical system. There are many ways to reduce the order of a dynamical system. However, we shall study only one of them: the balanced truncation method. The main advantage of this method is that it is simple and performs fairly well. A model order-reduction problem can, in general, be stated as follows: Given a fullorder model G(s), find a lower-order model (say, an rth order model Gr ), such that G and Gr are close in some sense. Of course, there are many ways to define the closeness of an approximation. For example, one may desire that the reduced model be such that G = Gr + ∆a and ∆a is small in some norm. This model reduction is usually called an additive model reduction problem. We shall be only interested in L∞ norm approximation in this book. Once the norm is chosen, the additive model reduction problem can be formulated as inf deg(Gr )≤r kG − Gr k∞ . In general, a practical model reduction problem is inherently frequency-weighted (i.e., the requirement on the approximation accuracy at one frequency range can be drastically different from the requirement at another frequency range). These problems can, in general, be formulated as frequency-weighted model reduction problems: inf deg(Gr )≤r kWo (G − Gr )Wi k∞ 105 106 BALANCED MODEL REDUCTION with an appropriate choice of Wi and Wo . We shall see in this chapter how the balanced realization can give an effective approach to the aforementioned model reduction problems. 7.1 Lyapunov Equations Testing stability, controllability, and observability of a system is very important in linear system analysis and synthesis. However, these tests often have to be done indirectly. In that respect, the Lyapunov theory is sometimes useful. Consider the following Lyapunov equation: A∗ Q + QA + H = 0 (7.1) with given real matrices A and H. It is well known that this equation has a unique solution iff λi (A) + λ̄j (A) 6= 0, ∀i, j. In this section, we shall study the relationships between the solution Q and the stability of A. The following results are standard. Lemma 7.1 Assume that A is stable, then the following statements hold: R∞ ∗ (i) Q = 0 eA t HeAt dt. (ii) Q > 0 if H > 0 and Q ≥ 0 if H ≥ 0. (iii) If H ≥ 0, then (H, A) is observable iff Q > 0. An immediate consequence of part (iii) is that, given a stable matrix A, a pair (C, A) is observable if and only if the solution to the following Lyapunov equation A∗ Q + QA + C ∗ C = 0 is positive definite, where Q is the observability Gramian. controllable if and only if the solution to Similarly, a pair (A, B) is AP + P A∗ + BB ∗ = 0 is positive definite, where P is the controllability Gramian. In many applications, we are given the solution of the Lyapunov equation and need to conclude the stability of the matrix A. Lemma 7.2 Suppose Q is the solution of the Lyapunov equation (7.1), then (i) Reλi (A) ≤ 0 if Q > 0 and H ≥ 0. (ii) A is stable if Q > 0 and H > 0. (iii) A is stable if Q ≥ 0, H ≥ 0, and (H, A) is detectable. 7.2. Balanced Realizations 107 Proof. Let λ be an eigenvalue of A and v 6= 0 be a corresponding eigenvector, then Av = λv. Premultiply equation (7.1) by v ∗ and postmultiply equation (7.1) by v to get 2Re λ(v ∗ Qv) + v ∗ Hv = 0. Now if Q > 0, then v ∗ Qv > 0, and it is clear that Reλ ≤ 0 if H ≥ 0 and Reλ < 0 if H > 0. Hence (i) and (ii) hold. To see (iii), we assume Reλ ≥ 0. Then we must have v ∗ Hv = 0 (i.e., Hv = 0). This implies that λ is an unstable and unobservable mode, which contradicts the assumption that (H, A) is detectable. ✷ 7.2 Balanced Realizations Although there are infinitely many different state-space realizations for a given transfer matrix, some particular realizations have proven to be very useful in control engineering and signal processing. Here we will only introduce one class of realizations for stable transfer matrices that are most useful in control applications. To motivate the class of realizations, we first consider some simple facts.   A B be a state-space realization of a (not necessarily stable) Lemma 7.3 Let C D transfer matrix G(s). Suppose that there exists a symmetric matrix   P1 0 P = P∗ = 0 0 with P1 nonsingular such that AP + P A∗ + BB ∗ = 0. Now partition the realization (A, B, C, D) compatibly with P as   A11 A12 B1  A21 A22 B2  . C1 C2 D  A11 C1 is stable. Then B1 D  is also a realization of G. Moreover, (A11 , B1 ) is controllable if A11 Proof. Use the partitioned P and (A, B, C) to get  A11 P1 + P1 A∗11 + B1 B1∗ ∗ ∗ 0 = AP + P A + BB = A21 P1 + B2 B1∗ P1 A∗21 + B1 B2∗ B2 B2∗  , 108 BALANCED MODEL REDUCTION which gives B2 = 0 and A21 = 0 since is not controllable:    A11 A12 B1  A21 A22 B2  =  C1 C2 D P1 is nonsingular. Hence, part of the realization A11 0 C1 A12 A22 C2   B1 A11 0 = C1 D B1 D  . Finally, it follows from Lemma 7.1 that (A11 , B1 ) is controllable if A11 is stable. ✷ We also have the following:   A B be a state-space realization of a (not necessarily stable) Lemma 7.4 Let C D transfer matrix G(s). Suppose that there exists a symmetric matrix   Q1 0 ∗ Q=Q = 0 0 with Q1 nonsingular such that QA + A∗ Q + C ∗ C = 0. Now partition the realization (A, B, C, D) compatibly with Q as   A11 A12 B1  A21 A22 B2  . C1 C2 D   A11 B1 Then is also a realization of G. Moreover, (C1 , A11 ) is observable if A11 C1 D is stable. The preceding two lemmas suggest that to obtain a minimal realization from a stable nonminimal realization, one only needs to eliminate all states corresponding to the zero block diagonal term of the controllability Gramian P and the observability Gramian Q. In the case where P is not block diagonal, the following procedure can be used to eliminate noncontrollable subsystems:   A B be a stable realization. 1. Let G(s) = C D 2. Compute the controllability Gramian P ≥ 0 from AP + P A∗ + BB ∗ = 0.    Λ1 0   U1 3. Diagonalize P to get P = U1 U2 0 0   U1 U2 unitary. U2 ∗ with Λ1 > 0 and 7.2. Balanced Realizations 4. Then G(s) =  U1∗ AU1 CU1 109 U1∗ B D  is a controllable realization. A dual procedure can also be applied to eliminate nonobservable subsystems. Now assume that Λ1 > 0 is diagonal and is partitioned as Λ1 = diag(Λ11 , Λ12 ) such that λmax (Λ12 ) ≪ λmin (Λ11 ); then it is tempting to conclude that one can also discard those states corresponding to Λ12 without causing much error. However, this is not necessarily true, as shown in the following example. Example 7.1 Consider a stable transfer function G(s) = s2 3s + 18 . + 3s + 18 Then G(s) has a state-space realization given by   −1 −4/α 1 −2 2α  G(s) =  4α −1 2/α 0 where α is any nonzero number. It is easy to check that the controllability Gramian of the realization is given by   0.5 P = . α2 Since the last diagonal term of P can be made arbitrarily small by making α small, the controllability of the corresponding state can be made arbitrarily weak. If the state corresponding to the last diagonal term of P is removed, we get a transfer function   −1 −1 1 = , Ĝ = −1 0 s+1 which is not close to the original transfer function in any sense. The problem may be easily detected if one checks the observability Gramian Q, which is   0.5 Q= . 1/α2 Since 1/α2 is very large if α is small, this shows that the state corresponding to the last diagonal term is strongly observable. This example shows that the controllability (or observability) Gramian alone cannot give an accurate indication of the dominance of the system states in the input/output 110 BALANCED MODEL REDUCTION behavior. This motivates the introduction of a balanced realization that gives balanced Gramians for controllability and observability.  A B is stable (i.e., A is stable). Let P and Q denote the Suppose G = C D controllability Gramian and observability Gramian, respectively. Then by Lemma 7.1, P and Q satisfy the following Lyapunov equations: AP + P A∗ + BB ∗ = 0 (7.2) A∗ Q + QA + C ∗ C = 0, (7.3) and P ≥ 0, Q ≥ 0. Furthermore, the pair (A, B) is controllable iff P > 0, and (C, A) is observable iff Q > 0. Suppose the state is transformed by a nonsingular T to x̂ = T x to yield the realization #  "  Â B̂ T AT −1 T B = . G= CT −1 D Ĉ D̂ Then the Gramians are transformed to P̂ = T P T ∗ and Q̂ = (T −1 )∗ QT −1 . Note that P̂ Q̂ = T P QT −1, and therefore the eigenvalues of the product of the Gramians are invariant under state transformation. Consider the similarity transformation T , which gives the eigenvector decomposition P Q = T −1 ΛT, Λ = diag(λ1 Is1 , . . . , λN IsN ). Then the columns of T −1 are eigenvectors of P Q corresponding to the eigenvalues {λi }. Later, it will be shown that P Q has a real diagonal Jordan form and that Λ ≥ 0, which are consequences of P ≥ 0 and Q ≥ 0. Although the eigenvectors are not unique, in the case of a minimal realization they can always be chosen such that P̂ = T P T ∗ = Σ, Q̂ = (T −1 )∗ QT −1 = Σ, where Σ = diag(σ1 Is1 , σ2 Is2 , . . . , σN IsN ) and Σ2 = Λ. This new realization with controllability and observability Gramians P̂ = Q̂ = Σ will be referred to as a balanced realization (also called internally balanced realization). The decreasingly ordered numbers, σ1 > σ2 > . . . > σN ≥ 0, are called the Hankel singular values of the system. More generally, if a realization of a stable system is not minimal, then there is a transformation such that the controllability and observability Gramians for the transformed realization are diagonal and the controllable and observable subsystem is balanced. This is a consequence of the following matrix fact. 7.2. Balanced Realizations 111 Theorem 7.5 Let P and Q be two nonsingular matrix T such that  Σ1  Σ2 TPT∗ =   0 0 positive semidefinite matrices. Then there exists a    ,   (T −1 )∗ QT −1 =    Σ1 0 Σ3 0 respectively, with Σ1 , Σ2 , Σ3 diagonal and positive definite.    Proof. Since P is a positive semidefinite matrix, there exists a transformation T1 such that   I 0 T1 P T1∗ = 0 0 Now let (T1∗ )−1 QT1−1 =  Q11 Q∗12 Q12 Q22  and there exists a unitary matrix U1 such that   2 Σ1 0 , Σ1 > 0 U1 Q11 U1∗ = 0 0 Let (T2∗ )−1 and then =  (T2∗ )−1 (T1∗ )−1 QT1−1 (T2 )−1 But Q ≥ 0 implies Q̂122 = 0. So now let  giving (T3∗ )−1 =  U1 0   Σ21 = 0 Q̂∗121 0 0 Q̂∗122 I 0  0 0  I −Q̂∗121 Σ−2 1 (T3∗ )−1 (T2∗ )−1 (T1∗ )−1 QT1−1 (T2 )−1 (T3 )−1 Next find a unitary matrix U2 such that U2 (Q22 − 0 I Q̂∗121 Σ1−2 Q̂121 )U2∗ 0 I 0  Σ21  0 = 0 =  Σ3 0  Q̂121 Q̂122  Q22  0 0  0 0 −2 ∗ 0 Q22 − Q̂121 Σ1 Q̂121 0 0  , Σ3 > 0 112 BALANCED MODEL REDUCTION Define (T4∗ )−1 and let  −1/2 Σ1 = 0 0  0 0 I 0  0 U2 T = T4 T3 T2 T1 Then   TPT∗ =   with Σ2 = I.  Σ1 Σ2 0 0  ,    Σ1 0  (T ∗ )−1 QT −1 =   Σ3 0    ✷ Corollary 7.6 The product of two positive semidefinite matrices is similar to a positive semidefinite matrix. Proof. Let P and Q be any positive semidefinite matrices. Then it is easy to see that with the transformation given previously  2  Σ1 0 −1 T P QT = 0 0 ✷  Corollary 7.7 For any stable system G = " T AT −1 T B formation T such that G = CT −1 D servability Gramian Q given by   Σ1   Σ2 , P =   0 0 A C # B D  , there exists a nonsingular trans- has controllability Gramian P and ob-   Q=   Σ1 0 respectively, with Σ1 , Σ2 , Σ3 diagonal and positive definite. Σ3 0    7.2. Balanced Realizations 113  A B is a minimal realization, a balanced realization C D can be obtained through the following simplified procedure: In the special case where  1. Compute the controllability and observability Gramians P > 0, Q > 0. 2. Find a matrix R such that P = R∗ R. 3. Diagonalize RQR∗ to get RQR∗ = U Σ2 U ∗ . 4. Let T −1 ∗ −1/2 = R UΣ ∗ ∗ −1 . Then T P T = (T ) QT −1 = Σ and " T AT −1 TB CT −1 D # is balanced. Assume that the Hankel singular values of the system are decreasingly ordered so that Σ = diag(σ1 Is1 , σ2 Is2 , . . . , σN IsN ) with σ1 > σ2 > . . . > σN and suppose σr ≫ σr+1 for some r. Then the balanced realization implies that those states corresponding to the singular values of σr+1 , . . . , σN are less controllable and less observable than those states corresponding to σ1 , . . . , σr . Therefore, truncating those less controllable and less observable states will not lose much information about the system. Two other closely related realizations are called input normal realization with P = I and Q = Σ2 , and output normal realization with P = Σ2 and Q = I. Both realizations can be obtained easily from the balanced realization by a suitable scaling on the states. Next we shall derive some simple and useful bounds for the H∞ norm and the L1 norm of a stable system. Theorem 7.8 Suppose G(s) =  A C B 0  ∈ RH∞ is a balanced realization; that is, there exists Σ = diag(σ1 Is1 , σ2 Is2 , . . . , σN IsN ) ≥ 0 with σ1 > σ2 > . . . > σN ≥ 0, such that AΣ + ΣA∗ + BB ∗ = 0 Then σ1 ≤ kGk∞ ≤ where g(t) = CeAt B. Z 0 ∞ A∗ Σ + ΣA + C ∗ C = 0 kg(t)k dt ≤ 2 N X σi i=1 Remark 7.1 It should be clear that the inequalities stated in the theorem do not depend on a particular state-space realization of G(s). However, use of the balanced realization does make the proof simple. ✸ 114 BALANCED MODEL REDUCTION Proof. Let G(s) have the following state-space realization: ẋ = Ax + Bw z = Cx. (7.4) Assume without loss of generality that (A, B) is controllable and (C, A) is observable. Then Σ is nonsingular. Next, differentiate x(t)∗ Σ−1 x(t) along the solution of equation (7.4) for any given input w as follows: d ∗ −1 (x Σ x) = ẋ∗ Σ−1 x + x∗ Σ−1 ẋ = x∗ (A∗ Σ−1 + Σ−1 A)x + 2hw, B ∗ Σ−1 xi dt Using the equation involving controllability Gramian to substitute for A∗ Σ−1 + Σ−1 A and completion of the squares gives d ∗ −1 (x Σ x) = kwk2 − kw − B ∗ Σ−1 xk2 dt Integration from t = −∞ to t = 0 with x(−∞) = 0 and x(0) = x0 gives x∗0 Σ−1 x0 = kwk22 − kw − B ∗ Σ−1 xk22 ≤ kwk22 Let w = B ∗ Σ−1 x; then ẋ = (A + BB ∗ Σ−1 )x = −ΣA∗ Σ−1 x =⇒ x ∈ L2 (−∞, 0] =⇒ w ∈ L2 (−∞, 0] and  kwk22 x(0) = x0 = x∗0 Σ−1 x0 . inf w∈L2 (−∞,0] Given x(0) = x0 and w = 0 for t ≥ 0, the norm of z(t) = CeAt x0 can be found from Z ∞ Z ∞ ∗ 2 kz(t)k dt = x∗0 eA t C ∗ CeAt x0 dt = x∗0 Σx0 0 0 To show σ1 ≤ kGk∞ , note that kGk∞ ≥ qR ∞ kz(t)k2 dt kg ∗ wk2 −∞ q = sup = sup R∞ w∈L2 (−∞,∞) kwk2 w∈L2 (−∞,∞) kw(t)k2 dt −∞ sup w∈L2 (−∞,0] qR ∞ 0 qR 0 −∞ 2 kz(t)k dt 2 kw(t)k dt = sup x0 6=0 s x∗0 Σx0 = σ1 x∗0 Σ−1 x0 We shall now show the other inequalities. Since Z ∞ g(t)e−st dt, Re(s) > 0, G(s) := 0 7.2. Balanced Realizations 115 by the definition of H∞ norm, we have kGk∞ = sup Re(s)>0 ≤ sup Re(s)>0 Z ∞ ≤ 0 Z Z ∞ 0 ∞ g(t)e−st dt g(t)e−st dt 0 kg(t)k dt. To prove the last inequality, let ei be the ith unit vector and define     E1 = e1 · · · es1 , E2 = es1 +1 · · · es1 +s2 ,   EN = es1 +···+sN −1 +1 · · · es1 +···+sN . N X Then ..., Ei Ei∗ = I and i=1 Z 0 ∞ kg(t)k dt = ≤ ≤ ≤ Z ∞ CeAt/2 0 N X Ei Ei∗ eAt/2 B dt i=1 N Z ∞ X 0 i=1 N Z X i=1 N X i=1 ∞ CeAt/2 Ei Ei∗ eAt/2 B dt CeAt/2 Ei 0 sZ ∞ CeAt/2 Ei 0 Ei∗ eAt/2 B dt 2 sZ dt ∞ 0 2 Ei∗ eAt/2 B dt ≤ 2 N X σi i=1 where we have used Cauchy-Schwarz inequality and the following relations: Z ∞ Z ∞   2 ∗ λmax Ei∗ eA t/2 C ∗ CeAt/2 Ei dt = 2λmax (Ei∗ ΣEi ) = 2σi CeAt/2 Ei dt = 0 0 Z 0 ∞ Ei∗ eAt/2 B 2 dt = Z 0 ∞   ∗ λmax Ei∗ eAt/2 BB ∗ eA t/2 Ei dt = 2λmax (Ei∗ ΣEi ) = 2σi ✷ Example 7.2 Consider a system   −1 −2 1 0 0  G(s) =  1 2 3 0 116 BALANCED MODEL REDUCTION It is easy to show that the Hankel singular values of G are σ1 = 1.6061 and σ2 = 0.8561. The H∞ norm of G is kGk∞ = 2.972 and the L1 norm of g(t) can be computed as Z ∞ |g(t)|dt = h1 + h2 + h3 + h4 + . . . 0 where hi , i =R 1, 2, . . . are the variations of the step response of G shown in Figure 7.1, ∞ which gives 0 |g(t)|dt ≈ 3.5. (See Problem 7.2.) 2.5 2 step response h2 h4 1.5 h3 h1 1 0.5 0 0 1 2 3 4 5 time 6 7 8 9 10 Figure 7.1: Estimating the L1 norm of g(t) So we have 1.6061 = σ1 ≤ kGk∞ = 2.972 ≤ Z 0 ∞ |g(t)|dt = 3.5 ≤ 2(σ1 + σ2 ) = 4.9244. Illustrative MATLAB Commands: ≫ [Ab, Bb, Cb, sig, Tinv]=balreal(A, B, C); singular values and Tinv = T −1 ; % sig is a vector of Hankel ≫ [Gb , sig] = sysbal(G); Related MATLAB Commands: ssdelete, ssselect, modred, strunc 7.3. Model Reduction by Balanced Truncation 7.3 117 Model Reduction by Balanced Truncation  A B is a balanced realizaC D tion (i.e., its controllability and observability Gramians are equal and diagonal). Denote the balanced Gramians by Σ; then Consider a stable system G ∈ RH∞ and suppose G =  AΣ + ΣA∗ + BB ∗ = 0 (7.5) (7.6) A∗ Σ + ΣA + C ∗ C = 0.   Σ1 0 and partition the system Now partition the balanced Gramian as Σ = 0 Σ2 accordingly as   B1 A11 A12 B2  . G =  A21 A22 C1 C2 D The following theorem characterizes the properties of these subsystems. Theorem 7.9 Assume that Σ1 and Σ2 have no diagonal entries in common. Then both subsystems (Aii , Bi , Ci ), i = 1, 2 are asymptotically stable. Proof. It is clearly sufficient to show that A11 is asymptotically stable. The proof for the stability of A22 is similar. Note that equations (7.5) and (7.6) can be written in terms of their partitioned matrices as A11 Σ1 + Σ1 A∗11 + B1 B1∗ Σ1 A11 + A∗11 Σ1 + C1∗ C1 A21 Σ1 + Σ2 A∗12 + B2 B1∗ Σ2 A21 + A∗12 Σ1 + C2∗ C1 A22 Σ2 + Σ2 A∗22 + B2 B2∗ Σ2 A22 + A∗22 Σ2 + C2∗ C2 = 0 = = = = (7.7) 0 0 0 0 (7.8) (7.9) (7.10) (7.11) = 0. (7.12) By Lemma 7.3 or Lemma 7.4, Σ1 can be assumed to be positive definite without loss of generality. Then it is obvious that Reλi (A11 ) ≤ 0 by Lemma 7.2. Assume that A11 is not asymptotically stable; then there exists an eigenvalue at jω for some ω. Let V be a basis matrix for Ker(A11 − jωI). Then we have (A11 − jωI)V = 0, which gives V ∗ (A∗11 + jωI) = 0. (7.13) 118 BALANCED MODEL REDUCTION Equations (7.7) and (7.8) can be rewritten as (A11 − jωI)Σ1 + Σ1 (A∗11 + jωI) + B1 B1∗ = 0 Σ1 (A11 − jωI) + (A∗11 + jωI)Σ1 + C1∗ C1 = 0. (7.14) (7.15) Multiplication of equation (7.15) from the right by V and from the left by V ∗ gives V ∗ C1∗ C1 V = 0, which is equivalent to C1 V = 0. Multiplication of equation (7.15) from the right by V now gives (A∗11 + jωI)Σ1 V = 0. Analogously, first multiply equation (7.14) from the right by Σ1 V and from the left by V ∗ Σ1 to obtain B1∗ Σ1 V = 0. Then multiply equation (7.14) from the right by Σ1 V to get (A11 − jωI)Σ21 V = 0. It follows that the columns of Σ21 V are in Ker(A11 − jωI). Therefore, there exists a matrix Σ̄1 such that Σ21 V = V Σ̄21 . Since Σ̄21 is the restriction of Σ21 to the space spanned by V , it follows that it is possible to choose V such that Σ̄21 is diagonal. It is then also possible to choose Σ̄1 diagonal and such that the diagonal entries of Σ̄1 are a subset of the diagonal entries of Σ1 . Multiply equation (7.9) from the right by Σ1 V and equation (7.10) by V to get A21 Σ21 V + Σ2 A∗12 Σ1 V Σ2 A21 V + A∗12 Σ1 V = 0 = 0, which gives (A21 V )Σ̄21 = Σ22 (A21 V ). This is a Sylvester equation in (A21 V ). Because Σ̄21 and Σ22 have no diagonal entries in common, it follows that A21 V = 0 (7.16) is the unique solution. Now equations (7.16) and (7.13) imply that      V A11 A12 V = jω , 0 0 A21 A22 which means that the A-matrix of the original system has an eigenvalue at jω. This contradicts the fact that the original system is asymptotically stable. Therefore, A11 must be asymptotically stable. ✷ 7.3. Model Reduction by Balanced Truncation 119 Corollary 7.10 If Σ has distinct singular values, then every subsystem is asymptotically stable. The stability condition in Theorem 7.9 is only sufficient as shown in the following example. Example 7.3 Note that  −2 −2 −2.8284 (s − 1)(s − 2)  0 −1 −1.4142  = (s + 1)(s + 2) 2 1.4142 1  is a balanced realization with Σ = I, and every subsystem of the realization is stable. On the other hand,   −1 1.4142 1.4142 s2 − s + 2   0 0 = −1.4142 s2 + s + 2 −1.4142 0 1 is also a balanced realization with Σ = I, but one of the subsystems is not stable. Theorem 7.11 Suppose G(s) ∈ RH∞ and  A11 A12 G(s) =  A21 A22 C1 C2 B1 B2 D is a balanced realization with Gramian Σ = diag(Σ1 , Σ2 ) Σ1 Σ2   = diag(σ1 Is1 , σ2 Is2 , . . . , σr Isr ) = diag(σr+1 Isr+1 , σr+2 Isr+2 , . . . , σN IsN ) and σ1 > σ2 > · · · > σr > σr+1 > σr+2 > · · · > σN where σi has multiplicity si , i = 1, 2, . . . , N and s1 + s2 + · · · + sN = n. Then the truncated system   A11 B1 Gr (s) = C1 D is balanced and asymptotically stable. Furthermore, kG(s) − Gr (s)k∞ ≤ 2(σr+1 + σr+2 + · · · + σN ) 120 BALANCED MODEL REDUCTION Proof. The stability of Gr follows from Theorem 7.9. We shall first show the one step model reduction. Hence we shall assume Σ2 = σN IsN . Define the approximation error     B1 A11 A12 A11 B1   B2 A21 A22 − E11 := C1 D C1 C2 D   A11 0 0 B1  0 B1  A A 11 12  =   0 A21 A22 B2  −C1 C1 C2 0 Apply a similarity transformation T to the preceding state-space realization with     I I 0 I/2 I/2 0 T =  I/2 −I/2 0  , T −1 =  I −I 0  0 0 I 0 0 I to get E11  A11  0 =   A21 0  A12 /2 B1 −A12 /2 0   A22 B2  C2 0 0 A11 −A21 −2C1 Consider a dilation of E11 (s):   E11 (s) E12 (s) E(s) = E21 (s) E22 (s)  A11 0  0 A 11  A −A =  21 21   0 −2C1 −2σN B1∗ Σ−1 0 1 " # Ã B̃ =: C̃ D̃ B1 A12 /2 −A12 /2 0 A22 B2 C2 0 2σN I −B2∗ Then it is easy to verify that  satisfies Σ1 P̃ =  0 0  0 2 −1 σN Σ1 0  0 2σN IsN ÃP̃ + P̃ Ã∗ + B̃ B̃ ∗ ∗ P̃ C̃ + B̃ D̃ ∗ = 0 = 0 0 ∗ σN Σ−1 1 C1 ∗ −C2 2σN I 0       7.3. Model Reduction by Balanced Truncation Using these two equations, we have  Ã  0 ∼ E(s)E (s) =  C̃  Ã  =  0 C̃  Ã  0 =  C̃ −B̃ B̃ ∗ −Ã∗ −D̃B̃ ∗  B̃ D̃∗ C̃ ∗   D̃D̃∗ −ÃP̃ − P̃ Ã∗ − B̃ B̃ ∗ −Ã∗ −C̃ P̃ − D̃ B̃ ∗  0 0 −Ã∗ C̃ ∗   0 D̃D̃∗ 121  P̃ C̃ ∗ + B̃ D̃∗  C̃ ∗  ∗ D̃D̃ 2 = D̃D̃∗ = 4σN I where the second equality is obtained by applying a similarity transformation   I P̃ T = 0 I Hence kE11 k∞ ≤ kEk∞ = 2σN , which is the desired result. The remainder of the proof isachieved byusing the order reduction by one-step reA11 B1 sults and by noting that Gk (s) = obtained by the “kth” order partitioning C1 D is internally balanced with balanced Gramian given by Σ1 = diag(σ1 Is1 , σ2 Is2 , . . . , σk Isk ) Let Ek (s) = Gk+1 (s) − Gk (s) for k = 1, 2, . . . , N − 1 and let GN (s) = G(s). Then σ [Ek (jω)] ≤ 2σk+1 since Gk (s) is a reduced-order model obtained from the internally balanced realization of Gk+1 (s) and the bound for one-step order reduction holds. Noting that N −1 X G(s) − Gr (s) = Ek (s) k=r by the definition of Ek (s), we have σ [G(jω) − Gr (jω)] ≤ N −1 X k=r σ [Ek (jω)] ≤ 2 N −1 X σk+1 k=r This is the desired upper bound. A useful consequence of the preceding theorem is the following corollary. ✷ 122 BALANCED MODEL REDUCTION Corollary 7.12 Let σi , i = 1, . . . , N be the Hankel singular values of G(s) ∈ RH∞ . Then kG(s) − G(∞)k∞ ≤ 2(σ1 + . . . + σN ) The above bound can be tight for some systems. Example 7.4 Consider an nth-order transfer function G(s) = n X i=1 bi , s + ai P with ai > 0 and bi > 0. Then kG(s)k∞ = G(0) = ni=1 bi /ai state-space realization:  √ −a1 √b1  −a2 b2   .. ..  G= . √.  −a bn n  √ √ √ b1 b2 · · · bn 0 and G(s) has the following         and the controllability and observability Gramians of the realization are given by "p # bi bj P =Q= ai + aj It is easy to see that σi = λi (P ) = λi (Q) and n X n X n X 1 1 bi = G(0) = kGk∞ σi = λi (P ) = trace(P ) = 2a 2 2 i i=1 i=1 i=1 In particular, let ai = bi = α2i ; then P = Q → 12 In (i.e., σj → 21 as α → ∞). This example also shows that even when the Hankel singular values are extremely close, they may not be regarded as repeated singular values. The model reduction bound can also be loose for systems with Hankel singular values close to each other. 7.3. Model Reduction by Balanced Truncation 123 Example 7.5 Consider the balanced realization of a fourth-order system: G(s) =    =   (s − 0.99)(s − 2)(s − 3)(s − 4) (s + 1)(s + 2)(s + 3)(s + 4) −9.2e + 00 −5.7e + 00 5.7e + 00 −8.1e − 07 −2.7e + 00 6.4e − 01 −1.3e + 00 1.5e − 06 4.3e + 00 1.3e − 03 with Hankel singular values given by  −2.7e + 00 1.3e + 00 −4.3e + 00 −6.4e − 01 1.5e − 06 1.3e − 03   −7.9e − 01 7.1e − 01 −1.3e + 00   −7.1e − 01 −2.7e − 06 −2.3e − 03  1.3e + 00 −2.3e − 03 1.0e + 00 σ1 = 0.9998, σ2 = 0.9988, σ3 = 0.9963, σ4 = 0.9923. The approximation errors and the estimated bounds are listed in the following table. The table shows that the actual error for an rth-order approximation is almost the same as 2σr+1 , which would be the estimated bound if we regard σr+1 = σr+2 = · · · = σ4 . In general, it is not hard to construct an nth-order system so that the rth-order balanced model reduction error is approximately 2σr+1 but the error bound is arbitrarily close to 2(n − r)σr+1 . One method to construct such a system is as follows: Let G(s) be a stable all-pass function, that is, G∼ (s)G(s) = I. Then there is a balanced realization for G so that the controllability and observability Gramians are P = Q = I. Next, make a very small perturbation to the balanced realization, then the perturbed system has a balanced realization with distinct singular values and P = Q ≈ I. This perturbed system will have the desired properties. r kG − Gr k∞ P Bounds: 2 4i=r+1 σi 2σr+1 0 1.9997 7.9744 1.9996 1 1.9983 5.9748 1.9976 2 1.9933 3.9772 1.9926 3 1.9845 1.9845 1.9845 The balanced realization and truncation can be done using the following Matlab commands: ≫ [Gb , sig] = sysbal(G); % find a balanced realization Gb and the Hankel singular values sig. ≫ Gr = strunc(Gb , 2); % truncate to the second-order. Related MATLAB Commands: reordsys, resid, Hankmr 124 BALANCED MODEL REDUCTION 7.4 Frequency-Weighted Balanced Model Reduction This section considers the extension of the balanced truncation method to the frequencyweighted case. Given the original full-order model G ∈ RH∞ , the input weighting matrix Wi ∈ RH∞ , and the output weighting matrix Wo ∈ RH∞ , our objective is to find a lower-order model Gr such that kWo (G − Gr )Wi k∞ is made as small as possible. Assume that G, Wi , and Wo have the following state-space realizations:       Ai Bi Ao Bo A B , Wo = G= , Wi = C 0 Ci Di Co Do with A ∈ Rn×n . Note that there is no loss of generality in assuming D = G(∞) = 0 since otherwise it can be eliminated by replacing Gr with D + Gr . Now the state-space realization for the weighted transfer matrix is given by   A 0 BCi BDi # " Ā B̄  Bo C Ao 0  0   Wo GWi =  =: 0 0 Ai Bi  C̄ 0 Do C Co 0 0 Let P̄ and Q̄ be the solutions to the following Lyapunov equations: ĀP̄ + P̄ Ā∗ + B̄ B̄ ∗ Q̄Ā + Ā∗ Q̄ + C̄ ∗ C̄ = 0 = 0 (7.17) (7.18) Then the input weighted Gramian P and the output weighted Gramian Q are defined by         In In , Q := In 0 Q̄ P := In 0 P̄ 0 0 It can be shown easily that P and Q satisfy the following lower-order equations: ∗  ∗       A 0  Q Q∗12 BCi Ai Q12 Q22  P ∗ P12 A Bo C P12 P22 0 Ao +   + P ∗ P12 A Bo C A 0 P12 P22 0 Ao ∗  Q Q∗12 BCi Ai Q12 Q22 +   + BDi Bi C ∗ Do∗ Co∗ BDi Bi  C ∗ Do∗ Co∗ =0 (7.19) ∗ =0 (7.20) The computation can be further reduced if Wi = I or Wo = I. In the case of Wi = I, P can be obtained from P A∗ + AP + BB ∗ = 0 (7.21) 7.4. Frequency-Weighted Balanced Model Reduction 125 while in the case of Wo = I, Q can be obtained from QA + A∗ Q + C ∗ C = 0 (7.22) Now let T be a nonsingular matrix such that ∗ T P T = (T −1 ∗ ) QT −1 =  Σ1 Σ2  (i.e., balanced) with Σ1 = diag(σ1 Is1 , . . . , σr Isr ) and Σ2 = diag(σr+1 Isr+1 , . . . , σN IsN ) and partition the system accordingly as  " #  A A12 B1 11 T AT −1 T B =  A21 A22 B2  CT −1 0 C1 C2 0 Then a reduced-order model Gr is obtained as   A11 B1 Gr = C1 0 Unfortunately, there is generally no known a priori error bound for the approximation error and the reduced-order model Gr is not guaranteed to be stable either. A very special frequency-weighted model reduction problem is the relative error model reduction problem where the objective is to find a reduced-order model Gr so that Gr = G(I + ∆rel ) and k∆rel k∞ is made as small as possible. ∆rel is usually called the relative error. In the case where G is square and invertible, this problem can be simply formulated as min degGr ≤r G−1 (G − Gr ) ∞ . Of course, the dual approximation problem Gr = (I + ∆rel )G can be obtained by taking the transpose of G. It turns out that the approximation Gr obtained below also serves as a multiplicative approximation: G = Gr (I + ∆mul ) where ∆mul is usually called the multiplicative error. Error bounds can be derived if the frequency-weighted balanced truncation method is applied to the relative and multiplicative approximations. 126 BALANCED MODEL REDUCTION Theorem 7.13 Let G, G−1 ∈ RH∞ be an nth-order square transfer matrix with a state-space realization   A B G(s) = C D Let P and Q be the solutions to P A∗ + AP + BB ∗ = 0 Suppose Q(A − BD −1 C) + (A − BD −1 ∗ ∗ C) Q + C (D (7.23) −1 ∗ ) D −1 C=0 (7.24) P = Q = diag(σ1 Is1 , . . . , σr Isr , σr+1 Isr+1 , . . . , σN IsN ) = diag(Σ1 , Σ2 ) with σ1 > σ2 > . . . > σN ≥ 0, and let the realization of G be partitioned compatibly with Σ1 and Σ2 as   A11 A12 B1 G(s) =  A21 A22 B2  C1 C2 D Then   A11 B1 Gr (s) = C1 D is stable and minimum phase. Furthermore,  N  q Y 2 1 + 2σi ( 1 + σi + σi ) − 1 k∆rel k∞ ≤ i=r+1 k∆mul k∞ ≤ N  Y i=r+1  q 2 1 + 2σi ( 1 + σi + σi ) − 1 Related MATLAB Commands: srelbal, sfrwtbal 7.5 Notes and References Balanced realization was first introduced by Mullis and Roberts [1976] to study roundoff noise in digital filters. Moore [1981] proposed the balanced truncation method for modelreduction. The stability properties of the reduced-order model were shown by Pernebo and Silverman [1982]. The error bound for the balanced model reduction was shown by Enns [1984a, 1984b], and Glover [1984] subsequently gave an independent proof. The frequency-weighted balanced model-reduction method was also introduced by Enns [1984a, 1984b]. The error bounds for the relative error are derived in Zhou [1995]. Other related results are shown in Green [1988]. Other weighted model-reduction methods can be found in Al-Saggaf and Franklin [1988], Glover [1986b], Glover, Limebeer and Hung [1992], Green [1988], Hung and Glover [1986], Zhou [1995], and references therein. Discrete-time balance model-reduction results can be found in Al-Saggaf and Franklin [1987], Hinrichsen and Pritchard [1990], and references therein. 7.6. Problems 7.6 127 Problems Problem 7.1 Use the following relation to show that P = R∞ 0 ∗ ∗ d  At A∗ t  = AeAt QeA t + eAt QeA t A e Qe dt ∗ eAt QeA t dt solves AP + P A∗ + Q = 0 if A is stable. Problem 7.2 Let G(s) ∈ H∞ and let g(t) be the inverse Laplace transform of G(s). Let hi , i = 1, 2, . . . be the variations of the step response of G. Show that Z ∞ |g(t)|dt = h1 + h2 + h3 + h4 + . . . 0 Problem 7.3 Let Q ≥ 0 be the solution to QA + A∗ Q + C ∗ C = 0 Suppose Q has m zero eigenvalues. Show that there is a nonsingular matrix T such that  #  A " 0 B1 11 T AT −1 T B =  A21 A22 B2  , A22 ∈ Rm×m . CT −1 D C1 0 D Apply the above result to the following state-space model:      −4 −7 −2 1 2 0 2 0 0  , B =  0 −1  , C = A= 1 1 1 −1 1 0 0 2 Problem 7.4 Let G(s) = 5 X i=1 1 0  , D=0 α2i s + α2i Find a balanced realization for each of the following α: α = 2, 4, 20, 100. Discuss the behavior of the Hankel singular values as α → ∞. Problem 7.5 Find a transformation so that T P T ∗ = Σ2 , (T ∗ )−1 QT −1 = I. (This realization is called output normalized realization.) 128 BALANCED MODEL REDUCTION Problem 7.6 Consider the model reduction error:  A11 0 A12 /2 B1  0 0 A −A 11 12 /2 E11 =   A21 −A21 A22 B2 0 −2C1 C2 0 Show that satisfies  Σ1 P̃ =  0 0 0 2 −1 σN Σ1 0 Ae P̃ + P̃ A∗e + Be Be∗ +    =:   Ae Ce Be 0  .   0 2σN IsN 1 ∗ 2 P̃ Ce Ce P̃ = 0. 4σN Problem 7.7 Suppose P and Q are the controllability and observability Gramians of G(s) = C(sI − A)−1 B ∈ RH∞ . Let Gd (z) = G(s)|s= z+1 = Cd (zI − Ad )−1 Bd + Dd . z−1 Compute the controllability and observability Gramians Pd and Qd and compare P Q and Pd Qd . Problem 7.8 Note that a delay can be approximated as  τ n s 1 − 2n e−τ s ≈ τ s 1 + 2n e−s be approximated by 1 + Ts 10  1 1 − 0.05s G(s) = 1 + 0.05s 1 + sT for a sufficiently large n. Let a process model For each T = 0, 0.01, 0.1, 1, 10, find a reduced-order model, if possible, using balanced truncation such that the approximation error is no greater than 0.1. Chapter 8 Uncertainty and Robustness In this chapter we briefly describe various types of uncertainties that can arise in physical systems, and we single out “unstructured uncertainties” as generic errors that are associated with all design models. We obtain robust stability tests for systems under various model uncertainty assumptions through the use of the small gain theorem. We also obtain some sufficient conditions for robust performance under unstructured uncertainties. The difficulty associated with MIMO robust performance design and the role of plant condition numbers for systems with skewed performance and uncertainty specifications are revealed. A simple example is also used to indicate the fundamental difference between the robustness of an SISO system and that of a MIMO system. In particular, we show that applying the SISO analysis/design method to a MIMO system may lead to erroneous results. 8.1 Model Uncertainty Most control designs are based on the use of a design model. The relationship between models and the reality they represent is subtle and complex. A mathematical model provides a map from inputs to responses. The quality of a model depends on how closely its responses match those of the true plant. Since no single fixed model can respond exactly like the true plant, we need, at the very least, a set of maps. However, the modeling problem is much deeper — the universe of mathematical models from which a model set is chosen is distinct from the universe of physical systems. Therefore, a model set that includes the true physical plant can never be constructed. It is necessary for the engineer to make a leap of faith regarding the applicability of a particular design based on a mathematical model. To be practical, a design technique must help make this leap small by accounting for the inevitable inadequacy of models. A good model should be simple enough to facilitate design, yet complex enough to give the engineer confidence that designs based on the model will work on the true plant. The term uncertainty refers to the differences or errors between models and reality, 129 130 UNCERTAINTY AND ROBUSTNESS and whatever mechanism is used to express these errors will be called a representation of uncertainty. Representations of uncertainty vary primarily in terms of the amount of structure they contain. This reflects both our knowledge of the physical mechanisms that cause differences between the model and the plant and our ability to represent these mechanisms in a way that facilitates convenient manipulation. For example, consider the problem of bounding the magnitude of the effect of some uncertainty on the output of a nominally fixed linear system. A useful measure of uncertainty in this context is to provide a bound on the power spectrum of the output’s deviation from its nominal response. In the simplest case, this power spectrum is assumed to be independent of the input. This is equivalent to assuming that the uncertainty is generated by an additive noise signal with a bounded power spectrum; the uncertainty is represented as additive noise. Of course, no physical system is linear with additive noise, but some aspects of physical behavior are approximated quite well using this model. This type of uncertainty received a great deal of attention in the literature during the 1960s and 1970s, and elegant solutions are obtained for many interesting problems (e.g., white noise propagation in linear systems, Wiener and Kalman filtering, and LQG optimal control). Unfortunately, LQG optimal control did not address uncertainty adequately and hence had less practical impact than might have been hoped. Generally, the deviation’s power spectrum of the true output from the nominal will depend significantly on the input. For example, an additive noise model is entirely inappropriate for capturing uncertainty arising from variations in the material properties of physical plants. The actual construction of model sets for more general uncertainty can be quite difficult. For example, a set membership statement for the parameters of an otherwise known FDLTI model is a highly structured representation of uncertainty. It typically arises from the use of linear incremental models at various operating points (e.g., aerodynamic coefficients in flight control vary with flight environment and aircraft configurations, and equation coefficients in power plant control vary with aging, slag buildup, coal composition, etc.). In each case, the amounts of variation and any known relationships between parameters can be expressed by confining the parameters to appropriately defined subsets of parameter space. However, for certain classes of signals (e.g., high-frequency), the parameterized FDLTI model fails to describe the plant because the plant will always have dynamics that are not represented in the fixed order model. In general, we are forced to use not just a single parameterized model but model sets that allow for plant dynamics that are not explicitly represented in the model structure. A simple example of this involves using frequency domain bounds on transfer functions to describe a model set. To use such sets to describe physical systems, the bounds must roughly grow with frequency. In particular, at sufficiently high frequencies, phase is completely unknown (i.e., ±180o uncertainties). This is a consequence of dynamic properties that inevitably occur in physical systems. This gives a less structured representation of uncertainty. 8.1. Model Uncertainty 131 Examples of less structured representations of uncertainty are direct set membership statements for the transfer function matrix of the model. For instance, the statement P∆ (s) = P (s) + W1 (s)∆(s)W2 (s), σ[∆(jω)] < 1, ∀ω ≥ 0, (8.1) where W1 and W2 are stable transfer matrices that characterize the spatial and frequency structure of the uncertainty, confines the matrix P∆ to a neighborhood of the nominal model P . In particular, if W1 = I and W2 = w(s)I, where w(s) is a scalar function, then P∆ describes a disk centered at P with radius w(jω) at each frequency, as shown in Figure 8.1. The statement does not imply a mechanism or structure that gives rise to ∆. The uncertainty may be caused by parameter changes, as mentioned previously or by neglected dynamics, or by a host of other unspecified effects. An alternative statement to equation (8.1) is the so-called multiplicative form: P∆ (s) = (I + W1 (s)∆(s)W2 (s))P (s). (8.2) This statement confines P∆ to a normalized neighborhood of the nominal model P . An advantage of equation (8.2) over (8.1) is that in equation (8.2) compensated transfer functions have the same uncertainty representation as the raw model (i.e., the weighting functions apply to P K as well as P ). Some other alternative set membership statements will be discussed later. P(j ω ) w(jω ) Figure 8.1: Nyquist diagram of an uncertain model The best choice of uncertainty representation for a specific FDLTI model depends, of course, on the errors the model makes. In practice, it is generally possible to represent some of these errors in a highly structured parameterized form. These are usually the low-frequency error components. There are always remaining higher-frequency errors, however, which cannot be covered this way. These are caused by such effects as infinite-dimensional electromechanical resonance, time delays, diffusion processes, etc. Fortunately, the less structured representations, such as equations (8.1) and (8.2), are well suited to represent this latter class of errors. Consequently, equations (8.1) and 132 UNCERTAINTY AND ROBUSTNESS (8.2) have become widely used “generic” uncertainty representations for FDLTI models. An important point is that the construction of the weighting matrices W1 and W2 for multivariable systems is not trivial. Motivated from these observations, we will focus for the moment on the multiplicative description of uncertainty. We will assume that P∆ in equation (8.2) remains a strictly proper FDLTI system for all ∆. More general perturbations (e.g., time varying, infinite dimensional, nonlinear) can also be covered by this set provided they are given appropriate “conic sector” interpretations via Parseval’s theorem. This connection is developed in [Safonov, 1980] and [Zames, 1966] and will not be pursued here. When used to represent the various high-frequency mechanisms mentioned previously, the weighting functions in equation (8.2) commonly have the properties illustrated in Figure 8.2. They are small (≪ 1) at low frequencies and increase to unity and above at higher frequencies. The growth with frequency inevitably occurs because phase uncertainties eventually exceed ±180 degrees and magnitude deviations eventually exceed the nominal transfer function magnitudes. Readers who are skeptical about this reality are encouraged to try a few experiments with physical devices. nominal model log ω actual model Figure 8.2: Typical behavior of multiplicative uncertainty: pδ (s) = [1 + w(s)δ(s)]p(s) Also note that the representation of uncertainty in equation (8.2) can be used to include perturbation effects that are in fact certain. A nonlinear element may be quite accurately modeled, but because our design techniques cannot effectively deal with the nonlinearity, it is treated as a conic sector nonlinearity.1 As another example, we may deliberately choose to ignore various known dynamic characteristics in order to achieve a simple nominal design model. One such instance is the model reduction process discussed in the last chapter. 1 See, for example, Safonov [1980] and Zames [1966]. 8.1. Model Uncertainty 133 Example 8.1 Let a dynamical system be described by
10 (2 + 0.2α)s2 + (2 + 0.3α + 0.4β)s + (1 + 0.2β) P (s, α, β) = , (s2 + 0.5s + 1)(s2 + 2s + 3)(s2 + 3s + 6) α, β ∈ [−1, 1] Then for each frequency, all possible frequency responses with varying parameters α and β are in a box, as shown in Figure 8.3. We can also obtain an unstructured uncertainty bound for this system. In fact, we have P (s, α, β) ∈ {P0 + W ∆ | k∆k ≤ 1} with P0 := P (s, 0, 0) and
10 0.2s2 + 0.7s + 0.2 W (s) = P (s, 1, 1) − P (s, 0, 0) = 2 (s + 0.5s + 1)(s2 + 2s + 3)(s2 + 3s + 6) The frequency response P0 + W ∆ is shown in Figure 8.3 as circles. 0.5 0 −0.5 −1 −1.5 −2 −2.5 −3 −3.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Figure 8.3: Nyquist diagram of uncertain system and disk covering Another way to bound the frequency response is to treat α and β as norm bounded uncertainties; that is, P (s, α, β) ∈ {P0 + W1 ∆1 + W2 ∆2 | k∆i k∞ ≤ 1} with P0 = P (s, 0, 0) and W1 = 10(0.2s2 + 0.3s) , (s2 + 0.5s + 1)(s2 + 2s + 3)(s2 + 3s + 6) 134 UNCERTAINTY AND ROBUSTNESS W2 = (s2 10(0.4s + 0.2) + 0.5s + 1)(s2 + 2s + 3)(s2 + 3s + 6) It is in fact easy to show that {P0 + W1 ∆1 + W2 ∆2 | k∆i k∞ ≤ 1} = {P0 + W ∆ | k∆k∞ ≤ 1} with |W | = |W1 | + |W2 |. The frequency response P0 + W ∆ is shown in Figure 8.4. This bounding is clearly more conservative. 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −3 −3.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Figure 8.4: A conservative covering Example 8.2 Consider a process control model ke−τ s , 4 ≤ k ≤ 9, 2 ≤ T ≤ 3, 1 ≤ τ ≤ 2. Ts + 1 Take the nominal model as 6.5 G0 (s) = (2.5s + 1)(1.5s + 1) G(s) = Then for each frequency, all possible frequency responses are in a box, as shown in Figure 8.5. To obtain an unstructured uncertainty bound for this system, plot the error ∆a (jω) = G(jω) − G0 (jω) for a set of parameters, as shown in Figure 8.6, and then use the Matlab command ginput to pick a set of upper-bound frequency responses and use fitmag to fit a stable and minimum phase transfer function to the upper-bound frequency responses. 8.1. Model Uncertainty 135 4 2 ω =2 Imaginary 0 ω = 0.01 ω =3 ω =1 ω = 0.05 −2 ω = 0.1 −4 ω = 0.8 ω = 0.2 ω = 0.5 −6 −8 −5 0 ω = 0.3 Real 5 10 Figure 8.5: Uncertain delay system and G0 ≫ mf= ginput(50) % pick 50 points: the first column of mf is the frequency points and the second column of mf is the corresponding magnitude responses. ≫ magg=vpck(mf(:,2),mf(:,1)); % pack them as a varying matrix. ≫ Wa =fitmag(magg); % choose the order of Wa online. A third-order Wa is sufficient for this example. ≫ [A,B,C,D]=unpck(Wa) % converting into state-space. ≫ [Z, P, K]=ss2zp(A,B,C,D) % converting into zero/pole/gain form. We get Wa (s) = 0.0376(s + 116.4808)(s + 7.4514)(s + 0.2674) (s + 1.2436)(s + 0.5575)(s + 4.9508) and the frequency response of Wa is also plotted in Figure 8.6. Similarly, define the multiplicative uncertainty G(s) − G0 (s) ∆m (s) := G0 (s) and a Wm can be found such that |∆m (jω)| ≤ |Wm (jω)|, as shown in Figure 8.7. A Wm is given by 2.8169(s + 0.212)(s2 + 2.6128s + 1.732) Wm = s2 + 2.2425s + 2.6319 136 UNCERTAINTY AND ROBUSTNESS 1 10 0 10 −1 10 −2 10 −2 10 −1 10 0 10 1 10 2 10 Figure 8.6: ∆a (dashed line) and a bound Wa (solid line) 3 10 2 10 1 10 0 10 −1 10 −2 10 −3 10 −4 10 −5 10 −2 10 −1 10 0 10 1 10 Figure 8.7: ∆m (dashed line) and a bound Wm (solid line) 2 10 8.2. Small Gain Theorem 137 Note that this Wm is not proper since G0 and G do not have the same relative degrees. To get a proper Wm , we need to choose a nominal model G0 having the same the relative order as that of G. The following terminologies are used in this book: Definition 8.1 Given the description of an uncertainty model set Π and a set of performance objectives, suppose P ∈ Π is the nominal design model and K is the resulting controller. Then the closed-loop feedback system is said to have Nominal Stability (NS): if K internally stabilizes the nominal model P . Robust Stability (RS): if K internally stabilizes every plant belonging to Π. Nominal Performance (NP): if the performance objectives are satisfied for the nominal plant P . Robust Performance (RP): if the performance objectives are satisfied for every plant belonging to Π. The nominal stability and performance can be easily checked using various standard techniques. The conditions for which the robust stability and robust performance are satisfied under various assumptions on the uncertainty set Π will be considered in the following sections. Related MATLAB Commands: magfit, drawmag, fitsys, genphase, vunpck, vabs, vinv, vimag, vreal, vcjt, vebe 8.2 Small Gain Theorem This section and the next section consider the stability test of a nominally stable system under unstructured perturbations. The basis for the robust stability criteria derived in the sequel is the so-called small gain theorem. Consider the interconnected system shown in Figure 8.8 with M (s) a stable p × q transfer matrix. Theorem 8.1 (Small Gain Theorem) Suppose M ∈ RH∞ and let γ > 0. Then the interconnected system shown in Figure 8.8 is well-posed and internally stable for all ∆(s) ∈ RH∞ with (a) k∆k∞ ≤ 1/γ if and only if kM (s)k∞ < γ (b) k∆k∞ < 1/γ if and only if kM (s)k∞ ≤ γ 138 UNCERTAINTY AND ROBUSTNESS w1 e1 ✲e + ✻ + ✲ ∆ M ✛ e2 + w2 ❄ e✛+ Figure 8.8: M − ∆ loop for stability analysis Proof. We shall only prove part (a). The proof for part (b) is similar. Without loss of generality, assume γ = 1. (Sufficiency) It is clear that M (s)∆(s) is stable since both M (s) and ∆(s) are stable. Thus by Theorem 5.5 (or Corollary 5.4) the closed-loop system is stable if det(I − M ∆) has no zero in the closed right-half plane for all ∆ ∈ RH∞ and k∆k∞ ≤ 1. Equivalently, the closed-loop system is stable if inf σ (I − M (s)∆(s)) 6= 0 s∈C+ for all ∆ ∈ RH∞ and k∆k∞ ≤ 1. But this follows from inf σ (I − M (s)∆(s)) ≥ 1− sup σ̄ (M (s)∆(s)) = 1−kM (s)∆(s)k∞ ≥ 1−kM (s)k∞ > 0. s∈C+ s∈C+ (Necessity) This will be shown by contradiction. Suppose kM k∞ ≥ 1. We will show that there exists a ∆ ∈ RH∞ with k∆k∞ ≤ 1 such that det(I − M (s)∆(s)) has a zero on the imaginary axis, so the system is unstable. Suppose ω0 ∈ R+ ∪ {∞} is such that σ̄(M (jω0 )) ≥ 1. Let M (jω0 ) = U (jω0 )Σ(jω0 )V ∗ (jω0 ) be a singular value decomposition with   U (jω0 ) = u1 u2 · · · up   V (jω0 ) = v1 v2 · · · vq   σ1   σ2 Σ(jω0 ) =   .. . To obtain a contradiction, it now suffices to construct a ∆ ∈ RH∞ such that ∆(jω0 ) = 1 ∗ σ1 v1 u1 and k∆k∞ ≤ 1. Indeed, for such ∆(s), det(I − M (jω0 )∆(jω0 )) = det(I − U ΣV ∗ v1 u∗1 /σ1 ) = 1 − u∗1 U ΣV ∗ v1 /σ1 = 0 and thus the closed-loop system is either not well-posed (if ω0 = ∞) or unstable (if ω ∈ R). There are two different cases: 8.2. Small Gain Theorem 139 (1) ω0 = 0 or ∞: then U and V are real matrices. In this case, ∆(s) can be chosen as ∆= 1 v1 u∗1 ∈ Rq×p σ1 (2) 0 < ω0 < ∞: write u1 and v1 in the following form: u∗1 =  u11 ejθ1 u12 ejθ2 · · · u1p ejθp  ,    v1 =   v11 ejφ1 v12 ejφ2 .. . v1q ejφq      where u1i ∈ R and v1j ∈ R are chosen so that θi , φj ∈ [−π, 0) for all i, j. Choose βi ≥ 0 and αj ≥ 0 so that   βi − jω0 = θi , ∠ βi + jω0 ∠  αj − jω0 αj + jω0  = φj for i = 1, 2, . . . , p and j = 1, 2, . . . , q. Let  1 −s v11 α α1 +s h 1  ..  β −s ∆(s) =  u11 β11 +s  . σ1 αq −s v1q αq +s  Then k∆k∞ = 1/σ1 ≤ 1 and ∆(jω0 ) = β −s · · · u1p βpp +s i ∈ RH∞ 1 ∗ σ1 v1 u1 . ✷ The theorem still holds even if ∆ and M are infinite dimensional. This is summarized as the following corollary. Corollary 8.2 The following statements are equivalent: (i) The system is well-posed and internally stable for all ∆ ∈ H∞ with k∆k∞ < 1/γ; (ii) The system is well-posed and internally stable for all ∆ ∈ RH∞ with k∆k∞ < 1/γ; (iii) The system is well-posed and internally stable for all ∆ ∈ Cq×p with k∆k < 1/γ; (iv) kM k∞ ≤ γ. Remark 8.1 It can be shown that the small gain condition is sufficient to guarantee internal stability even if ∆ is a nonlinear and time-varying “stable” operator with an appropriately defined stability notion, see Desoer and Vidyasagar [1975]. ✸ 140 UNCERTAINTY AND ROBUSTNESS The following lemma shows that if kM k∞ > γ, there exists a destabilizing ∆ with k∆k∞ < 1/γ such that the closed-loop system has poles in the open right-half plane. (This is stronger than what is given in the proof of Theorem 8.1.) Lemma 8.3 Suppose M ∈ RH∞ and kM k∞ > γ. Then there exists a σ0 > 0 such that for any given σ ∈ [0, σ0 ] there exists a ∆ ∈ RH∞ with k∆k∞ < 1/γ such that det(I − M (s)∆(s)) has a zero on the axis Re(s) = σ. Proof. Without loss of generality, assume γ = 1. Since M ∈ RH∞ and kM k∞ > 1, there exists a 0 < ω0 < ∞ such that kM (jω0 )k > 1. Given any γ such that 1 < γ < kM (jω0 )k, there is a sufficiently small σ0 > 0 such that min kM (σ + jω0 )k ≥ γ σ∈[0,σ0 ] and s ω02 + (σ0 + α)2 ω02 + (σ0 − α)2 s ω02 + (σ0 + β)2 <γ ω02 + (σ0 − β)2 for any α ≥ 0 and β ≥ 0. Now let σ ∈ [0, σ0 ] and let M (σ + jω0 ) = U ΣV ∗ be a singular value decomposition with   U = u1 u2 · · · up   V = v1 v2 · · · vq   σ1   σ2 Σ= . .. . Write u1 and v1 in the following form: u∗1 =  u11 ejθ1 u12 ejθ2 · · · u1p ejθp  ,    v1 =   v11 ejφ1 v12 ejφ2 .. . v1q ejφq where u1i ∈ R and v1j ∈ R are chosen so that θi , φj ∈ [−π, 0) for all i, j. Choose βi ≥ 0 and αj ≥ 0 so that   βi − σ − jω0 ∠ = θi , βi + σ + jω0 ∠  αj − σ − jω0 αj + σ + jω0 for i = 1, 2, . . . , p and j = 1, 2, . . . , q. Let  α1 −s  α̃1 v11 α 1 +s h 1  ..  β −s ∆(s) =  β̃1 u11 β11 +s  . σ1 αq −s α̃q v1q αq +s       = φj β −s · · · β̃p u1p βpp +s i ∈ RH∞ 8.3. Stability under Unstructured Uncertainties where β̃i := Then k∆k∞ ≤ s ω02 + (σ + βi )2 , α̃j := ω02 + (σ − βi )2 n o maxj {α̃j } maxi β̃i σ1 ≤ and ∆(σ + jω0 ) = s 141 ω02 + (σ + αj )2 ω02 + (σ − αj )2 n o maxj {α̃j } maxi β̃i γ <1 1 v1 u∗1 σ1 det (I − M (σ + jω0 )∆(σ + jω0 )) = 0 Hence s = σ + jω0 is a zero for the transfer function det (I − M (s)∆(s)). ✷ The preceding lemma plays a key role in the necessity proofs of many robust stability tests in the sequel. 8.3 Stability under Unstructured Uncertainties The small gain theorem in the last section will be used here to derive robust stability tests under various assumptions of model uncertainties. The modeling error ∆ will again be assumed to be stable. (Most of the robust stability tests discussed in the sequel can be generalized easily to the unstable ∆ case with some mild assumptions on the number of unstable poles of the uncertain model; we encourage readers to fill in the details.) In addition, we assume that the modeling error ∆ is suitably scaled with weighting functions W1 and W2 (i.e., the uncertainty can be represented as W1 ∆W2 ). ✲f ✻ − ✲ K ✲ Π ✲ Figure 8.9: Unstructured robust stability analysis We shall consider the standard setup shown in Figure 8.9, where Π is the set of uncertain plants with P ∈ Π as the nominal plant and with K as the internally stabilizing controller for P . The sensitivity and complementary sensitivity matrix functions are defined, as usual, as So = (I + P K)−1 , To = I − So and Si = (I + KP )−1 , Ti = I − Si . 142 UNCERTAINTY AND ROBUSTNESS Recall that the closed-loop system is well-posed and internally stable if and only if  I K −Π I −1 =  (I + KΠ)−1 −K(I + ΠK)−1 (I + ΠK)−1 Π (I + ΠK)−1  ∈ RH∞ for all Π ∈ Π. 8.3.1 Additive Uncertainty We assume that the model uncertainty can be represented by an additive perturbation: Π = P + W1 ∆W2 . Theorem 8.4 Let Π = {P + W1 ∆W2 : ∆ ∈ RH∞ } and let K be a stabilizing controller for the nominal plant P . Then the closed-loop system is well-posed and internally stable for all k∆k∞ < 1 if and only if kW2 KSo W1 k∞ ≤ 1. Proof. Let Π = P + W1 ∆W2 ∈ Π. Then  =  I K −Π I −1 (I + KSo W1 ∆W2 )−1 Si (I + So W1 ∆W2 K)−1 So (P + W1 ∆W2 ) −KSo(I + W1 ∆W2 KSo )−1 So (I + W1 ∆W2 KSo )−1  is well-posed and internally stable if (I + ∆W2 KSo W1 )−1 ∈ RH∞ since det(I + KSo W1 ∆W2 ) = det(I + W1 ∆W2 KSo ) = det(I + So W1 ∆W2 K) = det(I + ∆W2 KSo W1 ). But (I + ∆W2 KSo W1 )−1 ∈ RH∞ is guaranteed if k∆W2 KSo W1 k∞ < 1 (small gain theorem). Hence kW2 KSo W1 k∞ ≤ 1 is sufficient for robust stability. To show the necessity, note that robust stability implies that K(I + ΠK)−1 = KSo (I + W1 ∆W2 KSo )−1 ∈ RH∞ for all admissible ∆. This, in turn, implies that ∆W2 K(I + ΠK)−1 W1 = I − (I + ∆W2 KSo W1 )−1 ∈ RH∞ for all admissible ∆. By the small gain theorem, this is true for all ∆ ∈ RH∞ with k∆k∞ < 1 only if kW2 KSo W1 k∞ ≤ 1. ✷ 8.3. Stability under Unstructured Uncertainties 143 d˜ ❄ Wd dm z w ✲ W2 ✲ ∆ ✲ ❄ e✲ W1 r ✲ e✲ K − ✻ d ❄ ❄ ✲ e ✲ e y✲ We ✲ P e✲ Figure 8.10: Output multiplicative perturbed systems 8.3.2 Multiplicative Uncertainty In this section, we assume that the system model is described by the following set of multiplicative perturbations: Π = (I + W1 ∆W2 )P with W1 , W2 , ∆ ∈ RH∞ . Consider the feedback system shown in Figure 8.10. Theorem 8.5 Let Π = {(I + W1 ∆W2 )P : ∆ ∈ RH∞ } and let K be a stabilizing controller for the nominal plant P . Then the closed-loop system is well-posed and internally stable for all ∆ ∈ RH∞ with k∆k∞ < 1 if and only if kW2 To W1 k∞ ≤ 1. Proof. We shall first prove that the condition is necessary for robust stability. Suppose kW2 To W1 k∞ > 1. Then by Lemma 8.3, for any given sufficiently small σ > 0, there is a ∆ ∈ RH∞ with k∆k∞ < 1 such that (I + ∆W2 To W1 )−1 has poles on the axis Re(s) = σ. This implies that (I + ΠK)−1 = So (I + W1 ∆W2 To )−1 has poles on the axis Re(s) = σ since σ can always be chosen so that the unstable poles are not cancelled by the zeros of So . Hence kW2 To W1 k∞ ≤ 1 is necessary for robust stability. The sufficiency follows from the small gain theorem. ✷ 144 8.3.3 UNCERTAINTY AND ROBUSTNESS Coprime Factor Uncertainty As another example, consider a left coprime factor perturbed plant described in Figure 8.11. z1 r ✲e ✲ K −✻ ✲∆ ˜N − ✲ e✛ ✲ Ñ w ✲❄ e ˜M ✛ ∆ z2 ✲ M̃ −1 y ✲ Figure 8.11: Left coprime factor perturbed systems Theorem 8.6 Let ˜ M )−1 (Ñ + ∆ ˜ N) Π = (M̃ + ∆ ˜M, ∆ ˜ N ∈ RH∞ . The transfer matrices (M̃ , Ñ ) are assumed to be a stable with M̃ , Ñ, ∆ left coprime factorization of P (i.e., P = M̃−1 Ñ ), and K internally stabilizes the  ˜ M . Then the closed-loop system is well˜ nominal system P . Define ∆ := ∆N ∆ posed and internally stable for all k∆k∞ < 1 if and only if   K (I + P K)−1 M̃ −1 ≤1 I ∞ Proof. Let K = U V −1 be a right coprime factorization over RH∞ . By Lemma 5.7, the closed-loop system is internally stable if and only if  ˜ N )U + (M̃ + ∆ ˜ M )V (Ñ + ∆ −1 ∈ RH∞ (8.3) Since K stabilizes P , (Ñ U + M̃ V )−1 ∈ RH∞ . Hence equation (8.3) holds if and only if −1  ˜ NU + ∆ ˜ M V )(Ñ U + M̃V )−1 ∈ RH∞ I + (∆ By the small gain theorem, the above is true for all k∆k∞ < 1 if and only if     U K (Ñ U + M̃V )−1 = (I + P K)−1 M̃ −1 ≤1 V I ∞ ∞ ✷ 8.3. Stability under Unstructured Uncertainties 8.3.4 145 Unstructured Robust Stability Tests Table 8.1 summaries robust stability tests on the plant uncertainties under various assumptions. All of the tests pertain to the standard setup shown in Figure 8.9, where Π is the set of uncertain plants with P ∈ Π as the nominal plant and with K as the internally stabilizing controller of P . W1 ∈ RH∞ W2 ∈ RH∞ ∆ ∈ RH∞ k∆k∞ < 1 Perturbed Model Sets Π (I + W1 ∆W2 )−1 P Representative Types of Uncertainty Characterized output (sensor) errors neglected HF dynamics uncertain rhp zeros input (actuators) errors neglected HF dynamics uncertain rhp zeros LF parameter errors uncertain rhp poles P (I + W1 ∆W2 )−1 LF parameter errors uncertain rhp poles (I + W1 ∆W2 )P P (I + W1 ∆W2 ) P (I + W1 ∆W2 P )−1 additive plant errors neglected HF dynamics uncertain rhp zeros LF parameter errors uncertain rhp poles ˜ M )−1 (Ñ + ∆ ˜ N) (M̃ + ∆ LF parameter errors P + W1 ∆W2 −1 P = M̃ Ñ   ˜M ˜N ∆ ∆= ∆ neglected HF dynamics uncertain rhp poles & zeros −1 (N + ∆N )(M + ∆M ) −1 P =N M  ∆N ∆= ∆M LF parameter errors neglected HF dynamics Robust Stability Tests kW2 To W1 k∞ ≤ 1 kW2 Ti W1 k∞ ≤ 1 kW2 So W1 k∞ ≤ 1 kW2 Si W1 k∞ ≤ 1 kW2 KSo W1 k∞ ≤ 1 kW2 So P W1 k∞ ≤ 1  K I  ≤1 So M̃ −1 M −1 Si [K I] ∞ ∞ ≤1 uncertain rhp poles & zeros Table 8.1: Unstructured robust stability tests (HF: high frequency, LF: low frequency) 146 UNCERTAINTY AND ROBUSTNESS Table 8.1 should be interpreted as follows: UNSTRUCTURED ANALYSIS THEOREM Given NS & Perturbed Model Sets Then Closed-Loop Robust Stability if and only if Robust Stability Tests The table also indicates representative types of physical uncertainties that can be usefully represented by cone-bounded perturbations inserted at appropriate locations. For example, the representation P∆ = (I + W1 ∆W2 )P in the first row is useful for output errors at high frequencies (HF), covering such things as unmodeled high-frequency dynamics of sensors or plants, including diffusion processes, transport lags, electromechanical resonances, etc. The representation P∆ = P (I + W1 ∆W2 ) in the second row covers similar types of errors at the inputs. Both cases should be contrasted with the third and the fourth rows, which treat P (I + W1 ∆W2 )−1 and (I + W1 ∆W2 )−1 P . These representations are more useful for variations in modeled dynamics, such as lowfrequency (LF) errors produced by parameter variations with operating conditions, with aging, or across production copies of the same plant. Note from the table that the stability requirements on ∆ do not limit our ability to represent variations in either the number or locations of rhp singularities, as can be seen from some simple examples. Example 8.3 Suppose an uncertain system with changing numbers of right-half plane poles is described by   1 P∆ = : δ ∈ R, |δ| ≤ 1 . s−δ 1 1 ∈ P∆ has one right-half plane pole and P2 = ∈ P∆ has no s−1 s+1 right-half plane pole. Nevertheless, the set of P∆ can be covered by a set of feedback uncertain plants:  P∆ ⊂ Π := P (1 + δP )−1 : δ ∈ RH∞ , kδk∞ ≤ 1 Then P1 = with P = 1 . s 8.4. Robust Performance 147 Example 8.4 As another example, consider the following set of plants: P∆ = s+1+α , |α| ≤ 2. (s + 1)(s + 2) This set of plants has changing numbers of right-half plane zeros since the plant has no right-half plane zero when α = 0 and has one right-half plane zero when α = −2. The uncertain plant can be covered by a set of multiplicative perturbed plants:   1 2δ (1 + ), δ ∈ RH∞ , kδk∞ ≤ 1 . P∆ ⊂ Π := s+2 s+1 It should be noted that this covering can be quite conservative. 8.4 Robust Performance Consider the perturbed system shown in Figure 8.12 with the set of perturbed models described by a set Π. Suppose the weighting matrices Wd , We ∈ RH∞ and the perford˜ ❄ Wd r✲e −✻ ✲ K ✲ P∆ ∈ Π d ❄ ✲ e y✲ We e✲ Figure 8.12: Diagram for robust performance analysis mance criterion is to keep the error e as small as possible in some sense for all possible models belonging to the set Π. In general, the set Π can be either a parameterized set or an unstructured set such as those described in Table 8.1. The performance specifications are usually specified in terms of the magnitude of each component e in the time domain with respect to bounded disturbances, or, alternatively and more conveniently, some requirements on the closed-loop frequency response of the transfer matrix between d˜ and e (say, integral of square error or the magnitude of the steady-state error with respect to sinusoidal disturbances). The former design criterion leads to the so-called 148 UNCERTAINTY AND ROBUSTNESS L1 -optimal control framework and the latter leads to H2 and H∞ design frameworks, respectively. In this section, we will focus primarily on the H∞ performance objectives with unstructured model uncertainty descriptions. The performance under structured uncertainty will be considered in Chapter 10. Suppose the performance criterion is to keep the worst-case energy of the error e as small as possible over all d˜ of unit energy, for example, sup kek2 ≤ ǫ kd˜k2 ≤1 for some small ǫ. By scaling the error e (i.e., by properly selecting We ) we can assume without loss of generality that ǫ = 1. Let Ted˜ denote the transfer matrix between d˜ and e, then Ted̃ = We (I + P∆ K)−1 Wd , P∆ ∈ Π. (8.4) Then the robust performance criterion in this case can be described as requiring that the closed-loop system be robustly stable and that Ted̃ ∞ ≤ 1, ∀P∆ ∈ Π. (8.5) More specifically, an output multiplicatively perturbed system will be analyzed first. The analysis for other classes of models can be done analogously. The perturbed model can be described as Π := {(I + W1 ∆W2 )P : ∆ ∈ RH∞ , k∆k∞ < 1} (8.6) with W1 , W2 ∈ RH∞ . The explicit system diagram is as shown in Figure 8.10. For this class of models, we have Ted˜ = We So (I + W1 ∆W2 To )−1 Wd , and the robust performance is satisfied iff kW2 To W1 k∞ ≤ 1 and Ted̃ ∞ ≤ 1, ∀∆ ∈ RH∞ , k∆k∞ < 1. The exact analysis for this robust performance problem is not trivial and will be given in Chapter 10. However, some sufficient conditions are relatively easy to obtain by bounding these two inequalities, and they may shed some light on the nature of these problems. It will be assumed throughout that the controller K internally stabilizes the nominal plant P . Theorem 8.7 Suppose P∆ ∈ {(I + W1 ∆W2 )P : ∆ ∈ RH∞ , k∆k∞ < 1} and K internally stabilizes P . Then the system robust performance is guaranteed if either one of the following conditions is satisfied: 8.4. Robust Performance 149 (i) for each frequency ω σ(Wd )σ(We So ) + σ(W1 )σ(W2 To ) ≤ 1; (8.7) κ(W1−1 Wd )σ(We So Wd ) + σ(W2 To W1 ) ≤ 1 (8.8) (ii) for each frequency ω where W1 and Wd are assumed to be invertible and number. κ(W1−1 Wd ) is the condition Proof. It is obvious that both condition (8.7) and condition (8.8) guarantee that kW2 To W1 k∞ ≤ 1. So it is sufficient to show that Ted˜ ∞ ≤ 1, ∀∆ ∈ RH∞ , k∆k∞ < 1. Now for any frequency ω, it is easy to see that σ(Ted˜) ≤ σ(We So )σ[(I + W1 ∆W2 To )−1 ]σ(Wd ) σ(We So )σ(Wd ) σ(We So )σ(Wd ) ≤ = σ(I + W1 ∆W2 To ) 1 − σ(W1 ∆W2 To ) σ(We So )σ(Wd ) ≤ . 1 − σ(W1 )σ(W2 To )σ̄(∆) Hence condition (8.7) guarantees σ(Ted̃ ) ≤ 1 for all ∆ ∈ RH∞ with k∆k∞ < 1 at all frequencies. Similarly, suppose W1 and Wd are invertible; write Ted̃ = We So Wd (W1−1 Wd )−1 (I + ∆W2 To W1 )−1 (W1−1 Wd ), and then σ(We So Wd )κ(W1−1 Wd ) . 1 − σ(W2 To W1 )σ̄(∆) Hence by condition (8.8), σ(Ted̃ ) ≤ 1 is guaranteed for all ∆ ∈ RH∞ with k∆k∞ < 1 at all frequencies. ✷ σ(Ted˜) ≤ Remark 8.2 It is not hard to show that either one of the conditions in the theorem is also necessary for scalar valued systems. Remark 8.3 Suppose κ(W1−1 Wd ) ≈ 1 (weighting matrices satisfying this condition are usually called round weights). This is particularly the case if W1 = w1 (s)I and Wd = wd (s)I. Recall that σ(We So Wd ) ≤ 1 is the necessary and sufficient condition for nominal performance and that σ(W2 To W1 ) ≤ 1 is the necessary and sufficient condition for robust stability. Hence the condition (ii) in Theorem 8.7 is almost guaranteed by NP + RS (i.e., RP is almost guaranteed by NP + RS). Since RP implies NP + RS, we have NP + RS ≈ RP. (In contrast, such a conclusion cannot be drawn in the skewed case, which will be considered in the next section.) Since condition (ii) implies NP+RS, we can also conclude that condition (ii) is almost equivalent to RP (i.e., beside being sufficient, it is almost necessary). ✸ 150 UNCERTAINTY AND ROBUSTNESS Remark 8.4 Note that in light of the equivalence relation between the robust stability and nominal performance, it is reasonable to conjecture that the preceding robust performance problem is equivalent to the robust stability problem in Figure 8.9 with the uncertainty model set given by Π := (I + Wd ∆e We )−1 (I + W1 ∆W2 )P and k∆e k∞ < 1, k∆k∞ < 1, as shown in Figure 8.13. This conjecture is indeed true; however, the equivalent model uncertainty is structured, and the exact stability analysis for such systems is not trivial and will be studied in Chapter 10. ✸ d˜ ✲ W2 e✲ K − ✻ z ✲ ∆ ✲ P w ✲ W1 ❄ Wd ∆e ✛ ✲❄ e ✲❄ e ✲ We e Figure 8.13: Robust performance with unstructured uncertainty vs. robust stability with structured uncertainty Remark 8.5 Note that if W1 and Wd are invertible, then Ted˜ can also be written as  −1 . Ted̃ = We So Wd I + (W1−1 Wd )−1 ∆W2 To W1 (W1−1 Wd ) So another alternative sufficient condition for robust performance can be obtained as σ(We So Wd ) + κ(W1−1 Wd )σ(W2 To W1 ) ≤ 1. A similar situation also occurs in the skewed case below. We will not repeat all these variations. ✸ 8.5 Skewed Specifications We now consider the system with skewed specifications (i.e., the uncertainty and performance are not measured at the same location). For instance, the system performance is still measured in terms of output sensitivity, but the uncertainty model is in input multiplicative form: Π := {P (I + W1 ∆W2 ) : ∆ ∈ RH∞ , k∆k∞ < 1} . 8.5. Skewed Specifications z 151 ❄ W1 W2 ✻ e ✲ K − ✻ d˜ w ✲ ∆ ✲❄ e ✲ P ❄ Wd ✲❄ e ✲ We e ✲ Figure 8.14: Skewed problems The system block diagram is shown in Figure 8.14. For systems described by this class of models, the robust stability condition becomes kW2 Ti W1 k∞ ≤ 1, and the nominal performance condition becomes kWe So Wd k∞ ≤ 1. To consider the robust performance, let T̃ed˜ denote the transfer matrix from d˜ to e. Then T̃ed̃ = We So (I + P W1 ∆W2 KSo )−1 Wd −1  . = We So Wd I + (Wd−1 P W1 )∆(W2 Ti W1 )(Wd−1 P W1 )−1 The last equality follows if W1 , Wd , and P are invertible and, if W2 is invertible, can also be written as  −1 T̃ed̃ = We So Wd (W1−1 Wd )−1 I + (W1−1 P W1 )∆(W2 P −1 W2−1 )(W2 To W1 ) (W1−1 Wd ). Then the following results follow easily. Theorem 8.8 Suppose P∆ ∈ Π = {P (I + W1 ∆W2 ) : ∆ ∈ RH∞ , k∆k∞ < 1} and K internally stabilizes P . Assume that P, W1 , W2 , and Wd are square and invertible. Then the system robust performance is guaranteed if either one of the following conditions is satisfied: (i) for each frequency ω σ(We So Wd ) + κ(Wd−1 P W1 )σ(W2 Ti W1 ) ≤ 1; (8.9) (ii) for each frequency ω κ(W1−1 Wd )σ(We So Wd ) + σ(W1−1 P W1 )σ(W2 P −1 W2−1 )σ(W2 To W1 ) ≤ 1. (8.10) 152 UNCERTAINTY AND ROBUSTNESS Remark 8.6 If the appropriate invertibility conditions are not satisfied, then an alternative sufficient condition for robust performance can be given by σ(Wd )σ(We So ) + σ(P W1 )σ(W2 KSo ) ≤ 1. Similar to the previous case, there are many different variations of sufficient conditions although equation (8.10) may be the most useful one. ✸ Remark 8.7 It is important to note that in this case, the robust stability condition is given in terms of Li = KP while the nominal performance condition is given in terms of Lo = P K. These classes of problems are called skewed problems or problems with skewed specifications.2 Since, in general, P K 6= KP , the robust stability margin or tolerances for uncertainties at the plant input and output are generally not the same. ✸ Remark 8.8 It is also noted that the robust performance condition is related to the condition number of the weighted nominal model. So, in general, if the weighted nominal model is ill-conditioned at the range of critical frequencies, then the robust performance condition may be far more restrictive than the robust stability condition and the nominal performance condition together. For simplicity, assume W1 = I, Wd = I and W2 = wt I, where wt ∈ RH∞ is a scalar function. Further, P is assumed to be invertible. Then the robust performance condition (8.10) can be written as σ(We So ) + κ(P )σ(wt To ) ≤ 1, ∀ω. Comparing these conditions with those obtained for nonskewed problems shows that the condition related to robust stability is scaled by the condition number of the plant.3 Since κ(P ) ≥ 1, it is clear that the skewed specifications are much harder to satisfy if the plant is not well conditioned. This problem will be discussed in more detail in Section 10.3.3 of Chapter 10. ✸ Remark 8.9 Suppose K is invertible, then T̃ed̃ can be written as T̃ed˜ = We K −1 (I + Ti W1 ∆W2 )−1 Si KWd . Assume further that We = I, Wd = ws I, W2 = I, where ws ∈ RH∞ is a scalar function. Then a sufficient condition for robust performance is given by κ(K)σ(Si ws ) + σ(Ti W1 ) ≤ 1, ∀ω, with κ(K) := σ(K)σ(K −1 ). This is equivalent to treating the input multiplicative plant uncertainty as the output multiplicative controller uncertainty. ✸ 2 See Stein and Doyle [1991]. condition can be derived so that the condition related to nominal performance is scaled by the condition number. 3 Alternative 8.5. Skewed Specifications 153 The fact that the condition number appeared in the robust performance test for skewed problems can be given another interpretation by considering two sets of plants Π1 and Π2 , as shown in Figure 8.15 and below. Π1 Π2 := {P (I + wt ∆) : ∆ ∈ RH∞ , k∆k∞ < 1} := {(I + w̃t ∆)P : ∆ ∈ RH∞ , k∆k∞ < 1} . ✲ wt ✲ ∆ ✲ w̃t ✲ ∆ ✲❄ e ✲ P ✲ ✲❄ e ✲ ✲ P Figure 8.15: Converting input uncertainty to output uncertainty Assume that P is invertible; then Π2 ⊇ Π1 if since P (I + wt ∆) = (I + wt P ∆P −1 )P . |w̃t | ≥ |wt |κ(P ) ∀ω The condition number of a transfer matrix can be very high at high frequency, which may significantly limit the achievable performance. The example below, taken from the textbook by Franklin, Powell, and Workman [1990, page 788], shows that the condition number shown in Figure 8.16 may increase with the frequency:   −0.2 0.1 1 0 1  −0.05 0   0 0 0.7    s (s + 1)(s + 0.07) = 1 0 0 −1 1 0 P (s) =   a(s) −0.05 0.7(s + 1)(s + 0.13)   1 0 0 0 0  0 1 0 0 0 where a(s) = (s + 1)(s + 0.1707)(s + 0.02929). It is appropriate to point out that the skewed problem setup, although more complicated than that of the nonskewed problem, is particularly suitable for control system design. To be more specific, consider the transfer function from w and d˜ to z and e:     w z = G(s) ˜ e d where G(s) := =    −W2 KSo Wd We So Wd    K 0 (I + P K)−1 P I We −W2 Ti W1 We So P W1 −W2 0 I   W1 0 0 Wd  154 UNCERTAINTY AND ROBUSTNESS 10 2 condition number 10 3 10 1 0 10 -3 10 10 -2 10 -1 10 0 10 1 10 2 frequency Figure 8.16: Condition number κ(ω) = σ̄(P (jω))/σ(P (jω)) Then a suitable performance criterion is to make kG(s)k∞ small. Indeed, small kG(s)k∞ implies that Ti , KSo , So P and So are small in some suitable frequency ranges, which are the desired design specifications discussed in Section 6.1 of Chapter 6. It will be clear in Chapter 16 and Chapter 17 that the kGk∞ is related to the robust stability margin in the gap metric, ν-gap metric, and normalized coprime factor perturbations. Therefore, making kGk∞ small is a suitable design approach. 8.6 Classical Control for MIMO Systems In this section, we show through an example that the classical control theory may not be reliable when it is applied to MIMO system design. Consider a symmetric spinning body with torque inputs, T1 and T2 , along two orthogonal transverse axes, x and y, as shown in Figure 8.17. Assume that the angular velocity of the spinning body with respect to the z axis is constant, Ω. Assume further that the inertias of the spinning body with respect to the x, y, and z axes are I1 , I2 = I1 , and I3 , respectively. Denote by ω1 and ω2 the angular velocities of the body with respect to the x and y axes, respectively. Then the Euler’s equation of the spinning 8.6. Classical Control for MIMO Systems 155 body is given by I1 ω̇1 − ω2 Ω(I1 − I3 ) = T1 I1 ω̇2 − ω1 Ω(I3 − I1 ) = T2 z x y Figure 8.17: Spinning body Define  u1 u2  :=  T1 /I1 T2 /I1  , a := (1 − I3 /I1 )Ω. Then the system dynamical equations can be written as        0 a u1 ω̇1 ω1 = + −a 0 u2 ω̇2 ω2 Now suppose that the angular rates ω1 and ω2 are measured in scaled and rotated coordinates:         1 y1 1 a ω1 cos θ sin θ ω1 = = y2 ω2 −a 1 ω2 cos θ − sin θ cos θ where tan θ := a. (There is no specific physical meaning for the measurements of y1 and y2 but they are assumed here only for the convenience of discussion.) Then the transfer matrix for the spinning body can be computed as Y (s) = P (s)U (s) 156 UNCERTAINTY AND ROBUSTNESS with   1 s − a2 a(s + 1) P (s) = 2 s + a2 −a(s + 1) s − a2 Suppose the control law is chosen to be a unit feedback u = −y. Then the sensitivity function and the complementary sensitivity function are given by     1 1 s −a 1 a , T = P (I + P )−1 = S = (I + P )−1 = s+1 a s s + 1 −a 1 1 Note that each single loop has the open-loop transfer function as , so each loop has s 90o phase margin and ∞ gain margin. Suppose one loop transfer function is perturbed, as shown in Figure 8.18. w y1 ✛ δ ✛ z ✻ ✛ c❄ P ✛ y2 u1 ✛ c ✻ u2 c − ✛ ✻ − Figure 8.18: One-loop-at-a-time analysis Denote 1 z(s) = −T11 = − w(s) s+1 Then the maximum allowable perturbation is given by kδk∞ < 1 = 1, kT11 k∞ which is independent of a. Similarly the maximum allowable perturbation on the other loop is also 1 by symmetry. However, if both loops are perturbed at the same time, then the maximum allowable perturbation is much smaller, as shown next. Consider a multivariable perturbation, as shown in Figure 8.19; that is, P∆ = (I + ∆)P , with   δ11 δ12 ∈ RH∞ ∆= δ21 δ22 a 2 × 2 transfer matrix such that k∆k∞ < γ. Then by the small gain theorem, the system is robustly stable for every such ∆ iff γ≤ 1 1 =√ kT k∞ 1 + a2 (≪ 1 if a ≫ 1). 8.7. Notes and References y1 ✛ 157 d✛ ✻ ♦ δ11 ✛ ❙ d✛ ✻ g11 δ21 ✛ ✴ ❄ d✓ ✛ − r̃ ❄ d✛ 1 g12 ✛ δ12 ✛ y2 ✛ ✛ g21 ✛ δ22 ✛ ❄ d✛ g22 ✛ d✛r̃2 ✻ − Figure 8.19: Simultaneous perturbations In particular, consider ∆ = ∆d =  δ11 δ22  ∈ R2×2 . Then the closed-loop system is stable for every such ∆ iff det(I + T ∆d ) =
1 s2 + (2 + δ11 + δ22 )s + 1 + δ11 + δ22 + (1 + a2 )δ11 δ22 2 (s + 1) has no zero in the closed right-half plane. Hence the stability region is given by 2 + δ11 + δ22 1 + δ11 + δ22 + (1 + a2 )δ11 δ22 > 0 > 0. It is easy to see that the system is unstable with 1 δ11 = −δ22 = √ . 1 + a2 The stability region for a = 5 is drawn in Figure 8.20, which shows how checking the axis misses nearby regions of instability, and that for a >> 5, things just get that much worse. The hyperbola portion of the picture gets arbitrarily close to (0,0). This clearly shows that the analysis of a MIMO system using SISO methods can be misleading and can even give erroneous results. Hence an MIMO method has to be used. 8.7 Notes and References The small gain theorem was first presented by Zames [1966]. The book by Desoer and Vidyasagar [1975] contains an extensive treatment and applications of this theorem in 158 UNCERTAINTY AND ROBUSTNESS 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Figure 8.20: Stability region for a = 5 various forms. Robust stability conditions under various uncertainty assumptions are discussed in Doyle, Wall, and Stein [1982]. 8.8 Problems Problem 8.1 This problem shows that the stability margin is critically dependent on the type of perturbation. The setup is a unity-feedback loop with controller K(s) = 1 and plant Pnom (s) + ∆(s), where Pnom (s) = 10 . s2 + 0.2s + 1 1. Assume ∆(s) ∈ RH∞ . Compute the largest β such that the feedback system is internally stable for all k∆k∞ < β. 2. Repeat but with ∆ ∈ R. Problem 8.2 Let M ∈ Cp×q be a given complex matrix. Then it is shown in Qiu et al [1995] that I − ∆M is invertible for all ∆ ∈ Rq×p such that σ(∆) ≤ γ if and only if µ∆ (M ) < 1/γ, where   ReM −αℑM . µ∆ (M ) = inf σ2 α−1 ℑM ReM α∈(0,1] 8.8. Problems 159 It follows that (I − ∆M (s))−1 ∈ RH∞ for a given M (s) ∈ RH∞ and all ∆ ∈ Rq×p with σ(∆) ≤ γ if and only if supω µ∆ (M (jω)) < 1/γ. Write a Matlab program to compute µ∆ (M ) and apply it to the preceding problem. Ke−τ s and K ∈ [10, 12], τ ∈ [0, 0.5], T = 1. Find a nominal Ts + 1 and a weighting function W (s) ∈ RH∞ such that Problem 8.3 Let G(s) = model Go (s) ∈ RH∞ G(s) ∈ {Go (s) (1 + W (s)∆(s)) : ∆ ∈ H∞ , k∆k ≤ 1} . Problem 8.4 Think of an example of a physical system with the property that the number of unstable poles changes when the system undergoes a small change, for example, when a mass is perturbed slightly or the geometry is deformed slightly. Problem 8.5 Let X be a space of scalar-valued transfer functions. A function f (s) in X is a unit if 1/f (s) is in X . 1. Prove that the set of units in RH∞ is an open set, that is, if f is a unit, then (∃ǫ > 0) (∀g ∈ RH∞ ) kgk∞ < ǫ =⇒ f + g is a unit. (8.11) 2. Here is an application of the preceding fact. Consider the unity feedback system with controller k(s) and plant p(s), both SISO, with k(s) proper and p(s) strictly proper. Do coprime factorizations over RH∞ : p= np , mp k= nk . mk Then the feedback system is internally stable iff np nk + mp mk is a unit in RH∞ . Assume it is a unit. Perturb p(s) to p= np + ∆n , mp + ∆m ∆n , ∆m ∈ RH∞ . Show that internal stability is preserved if k∆n k∞ and k∆m k∞ are small enough. The conclusion is that internal stability is preserved if the perturbations are small enough in the H∞ norm. 3. Give an example of a unit f (s) in RH∞ such that equation (8.11) fails for the H2 norm, that is, such that (∀ǫ > 0) (∃g ∈ RH2 ) kgk2 < ǫ and f + g is not a unit. What is the significance of this fact concerning robust stability? Problem 8.6 Let ∆ and M be square constant matrices. Prove that the following three conditions are equivalent: 160 1. UNCERTAINTY AND ROBUSTNESS  I −M −∆ I  is invertible; 2. I − M ∆ is invertible; 3. I − ∆M is invertible. Problem 8.7 Consider the unity feedback system ✲ j ✲ K −✻ For  1  s G(s) =   1 s ✲ G ✲ 1  s    1 s design a proper controller K(s) to stabilize the feedback system internally. Now perturb G(s) to  1+ǫ 1   s s    , ǫ ∈ R.   1 1 s s Is the feedback system internally stable for all sufficiently small ǫ? 1 . Coms−2 pute by hand (i.e., without Matlab) a normalized coprime factorization of G(s). Considering perturbations ∆N and ∆M of the factors by hand the stability   of G(s), compute ∆N ∆M such that feedback staradius ǫ, that is, the least upper bound on ∞ bility is preserved. Problem 8.8 Consider the unity feedback system with K(s) = 3, G(s) = 1 Problem 8.9 Let a unit feedback system with a controller K(s) = and a nominal s s+1 plant model Po (s) = 2 . Construct a smallest destabilizing ∆ ∈ RH∞ in the s + 0.2s + 5 sense of k∆k∞ for each of the following cases: (a) P = Po + ∆; (b) P = Po (1 + W ∆) with W (s) = 0.2(s + 10) ; s + 50 8.8. Problems (c) P = 161  N + ∆n s+1 s2 + 0.2s + 5 ,N= ,M= , and ∆ = ∆n 2 2 M + ∆m (s + 2) (s + 2)  ∆m . Problem 8.10 This problem concerns the unity feedback system with controller K(s) and plant   1 1 2 . G(s) = s+1 3 4 1. Take K(s) = kI2 (k a real scalar) and find the range of k for internal stability. 2. Take K(s) =  k1 0 0 k2  (k1 , k2 real scalars) and find the region of (k1 , k2 ) in R2 for internal stability. Problem 8.11 (Kharitonov’s Theorem) Let a(s) be an interval polynomial + − + − + 2 a(s) = [a− 0 , a0 ] + [a1 , a1 ]s + [a2 , a2 ]s + · · · . Kharitonov’s theorem shows that a(s) is stable if and only if the following four Kharitonov polynomials are stable: − + 2 + 3 − 4 − 5 + 6 K1 (s) = a− 0 + a1 s + a2 s + a3 s + a4 s + a5 s + a6 s + · · · + + 2 − 3 − 4 + 5 + 6 K2 (s) = a− 0 + a1 s + a2 s + a3 s + a4 s + a5 s + a6 s + · · · + − 2 − 3 + 4 + 5 − 6 K3 (s) = a+ 0 + a1 s + a2 s + a3 s + a4 s + a5 s + a6 s + · · · − − 2 + 3 + 4 − 5 − 6 K4 (s) = a+ 0 + a1 s + a2 s + a3 s + a4 s + a5 s + a6 s + · · · + Let ai := (a− i + ai )/2 and let anom (s) = a0 + a1 s + a2 s2 + · · · . Find a least conservative W (s) such that a(s) ∈ {1 + W (s)∆(s) | k∆k∞ ≤ 1} . anom (s) Problem 8.12 One of the main tools in this chapter was the small-gain theorem. One way to state it is as follows: Define a transfer matrix F (s) in RH∞ to be contractive if kF k∞ ≤ 1 and strictly contractive if kF k∞ < 1. Then for the unity feedback system the small gain theorem is this: If K is contractive and G is strictly contractive, then the feedback system is stable. This problem concerns passivity and the passivity theorem. This is an important tool in the study of the stability of feedback systems, especially robotics, that is complementary to the small gain theorem. 162 UNCERTAINTY AND ROBUSTNESS Consider a system with a square transfer matrix F (s) in RH∞ . This is said to be passive if F (jω) + F (jω)∗ ≥ 0, ∀ω. Here, the symbol ≥ 0 means that the matrix is positive semidefinite. If the system is SISO, the condition is equivalent to Re F (jω) ≥ 0, ∀ω; that is, the Nyquist plot of F lies in the right-half plane. The system is strictly passive if F − ǫI is passive for some ǫ > 0. 1. Consider a mechanical system with input vector u(t) (forces and torques) and output vector y(t) (velocities) modeled by the equation M ẏ + Ky = u where M and K are symmetric, positive definite matrices. Show that this system is passive. 2. If F is passive, then (I + F )−1 ∈ RH∞ and (I + F )−1 (I − F ) is contractive; if F is strictly passive, then (I + F )−1 (I − F ) is strictly contractive. Prove these statements for the case that F is SISO. 3. Using the results so far, show (in the MIMO case) that the unity feedback system is stable if K is passive and G is strictly passive. Problem 8.13 Consider a SISO feedback system shown below with P = Po + W2 ∆2 . d ❄ W3 e ✲ K − ✻ ✲ P ✲❄ e z ✲ Assume that P0 and P have the same number of right-half plane poles, W2 is stable, and |Re{∆2 }| ≤ α, |ℑ{∆2 }| ≤ β. Derive the necessary and sufficient conditions for the feedback system to be robustly stable. 8.8. Problems 163  P11 P12 ∈ RH∞ be a two-by-two transfer matrix. Find P21 P22 sufficient (and necessary, if possible) conditions in each case so that Fu (P, ∆) is stable for all possible stable ∆ that satisfies the following conditions, respectively: Problem 8.14 Let P =  1. at each frequency Re∆(jω) ≥ 0, |∆(jω)| < α 2. at each frequency Re∆(jω)e±jθ ≥ 0, |∆(jω)| < α where θ ≥ 0. 3. at each frequency Re∆(jω) ≥ 0, ℑ∆(jω) ≥ 0, Re∆(jω) + ℑ∆(jω) < α Problem 8.15 Let P = (I + ∆W )P0 such that P and P0 have the same number of unstable poles for all admissible ∆, k∆k∞ < γ. Show that K robustly stabilizes P if and only if K stabilizes P0 and W P0 K(I + P0 K)−1 ∞ ≤ 1. Problem 8.16 Give appropriate generalizations of the preceding problem to other types of uncertainties. Problem 8.17 Let K = I and 1  s+1 P0 =  1 s+1  2  s+3  . 1 s+1 1. Let P = P0 + ∆ with k∆k∞ ≤ γ. Determine the largest γ for robust stability.   k1 ∈ R2×2 . Determine the stability region. 2. Let ∆ = k2 Problem 8.18 Repeat the preceding problem with  5s + 1 s−1  (s + 1)2 (s + 1)2 P0 =   −1 s−1 (s + 1)2 (s + 1)2   .  164 UNCERTAINTY AND ROBUSTNESS Chapter 9 Linear Fractional Transformation This chapter introduces a new matrix function: linear fractional transformation (LFT). We show that many interesting control problems can be formulated in an LFT framework and thus can be treated using the same technique. 9.1 Linear Fractional Transformations This section introduces the matrix linear fractional transformations. It is well known from the one-complex-variable function theory that a mapping F : C 7→ C of the form F (s) = a + bs c + ds with a, b, c, and d ∈ C is called a linear fractional transformation. In particular, if c 6= 0 then F (s) can also be written as F (s) = α + βs(1 − γs)−1 for some α, β and γ ∈ C. The linear fractional transformation described above for scalars can be generalized to the matrix case. Definition 9.1 Let M be a complex matrix partitioned as   M11 M12 ∈ C(p1 +p2 )×(q1 +q2 ) , M= M21 M22 and let ∆ℓ ∈ Cq2 ×p2 and ∆u ∈ Cq1 ×p1 be two other complex matrices. Then we can formally define a lower LFT with respect to ∆ℓ as the map Fℓ (M, •) : Cq2 ×p2 7→ Cp1 ×q1 165 166 LINEAR FRACTIONAL TRANSFORMATION with Fℓ (M, ∆ℓ ) := M11 + M12 ∆ℓ (I − M22 ∆ℓ )−1 M21 provided that the inverse (I − M22 ∆ℓ )−1 exists. We can also define an upper LFT with respect to ∆u as Fu (M, •) : Cq1 ×p1 7→ Cp2 ×q2 with Fu (M, ∆u ) = M22 + M21 ∆u (I − M11 ∆u )−1 M12 provided that the inverse (I − M11 ∆u )−1 exists. The matrix M in the preceding LFTs is called the coefficient matrix. The motivation for the terminologies of lower and upper LFTs should be clear from the following diagram representations of Fℓ (M, ∆ℓ ) and Fu (M, ∆u ): z1 ✛ M y1 ✲ ∆ℓ w1 ✛ ✛ ✲ ∆u y2 u1 z2 ✛ M u2 ✛ ✛ w2 The diagram on the left represents the following set of equations:        w1 M11 M12 w1 z1 , = = M u1 M21 M22 u1 y1 u1 = ∆ℓ y1 while the diagram on the right represents      M11 u2 y2 = = M M21 w2 z2 u2 = ∆u y2 . M12 M22  u2 w2  , It is easy to verify that the mapping defined on the left diagram is equal to Fℓ (M, ∆ℓ ) and the mapping defined on the right diagram is equal to Fu (M, ∆u ). So from the above diagrams, Fℓ (M, ∆ℓ ) is a transformation obtained from closing the lower loop on the left diagram; similarly, Fu (M, ∆u ) is a transformation obtained from closing the upper loop on the right diagram. In most cases, we shall use the general term LFT in referring to both upper and lower LFTs and assume that the context will distinguish the situations since one can use either of these notations  to express a given object. Indeed, it is clear M22 M21 that Fu (N, ∆) = Fℓ (M, ∆) with N = . It is usually not crucial which M12 M11 expression is used; however, it is often the case that one expression is more convenient than the other for a given problem. It should also be clear to the reader that in writing Fℓ (M, ∆) [or Fu (M, ∆)] it is implied that ∆ has compatible dimensions. 9.1. Linear Fractional Transformations 167 A useful interpretation of an LFT [e.g., Fℓ (M, ∆)] is that Fℓ (M, ∆) has a nominal mapping, M11 , and is perturbed by ∆, while M12 , M21 , and M22 reflect a prior knowledge as to how the perturbation affects the nominal map, M11 . A similar interpretation can be applied to Fu (M, ∆). This is why LFT is particularly useful in the study of perturbations, which is the focus of the next chapter. The physical meaning of an LFT in control science is obvious if we take M as a proper transfer matrix. In that case, the LFTs defined previously are simply the closed-loop transfer matrices from w1 7→ z1 and w2 7→ z2 , respectively; that is, Tzw1 = Fℓ (M, ∆ℓ ), Tzw2 = Fu (M, ∆u ) where M may be the controlled plant and ∆ may be either the system model uncertainties or the controllers. Definition 9.2 An LFT, Fℓ (M, ∆), is said to be well-defined (or well-posed) if (I − M22 ∆) is invertible. Note that this definition is consistent with the well-posedness definition of the feedback system, which requires that the corresponding transfer matrix be invertible in Rp (s). It is clear that the study of an LFT that is not well-defined is meaningless; hence throughout this book, whenever an LFT is invoked, it will be assumed implicitly that it is well-defined. It is also clear from the definition that, for any M , Fℓ (M, 0) is well-defined; hence any function that is not well-defined at the origin cannot be expressed as an LFT in terms of its variables. For example, f (δ) = 1/δ is not an LFT of δ. In some literature, LFT is used to refer to the following matrix functions: (A + BQ)(C + DQ)−1 (C + QD)−1 (A + QB) or where C is usually assumed to be invertible due to practical consideration. The following results follow from some simple algebra. Lemma 9.1 Suppose C is invertible. Then (A + BQ)(C + DQ)−1 = Fℓ (M, Q) (C + QD)−1 (A + QB) = Fℓ (N, Q) with M=  AC −1 C −1 B − AC −1 D −C −1 D  , N=  C −1 A C −1 −1 B − DC A −DC −1 The converse also holds if M satisfies certain conditions.  . 168 LINEAR FRACTIONAL TRANSFORMATION Lemma 9.2 Let Fℓ (M, Q) be a given LFT with M =  M11 M21 M12 M22  . (a) If M12 is invertible, then Fℓ (M, Q) = (C + QD)−1 (A + QB) −1 −1 with A = M12 M11 , B = M21 − M22 M12 M11 , that is,    0 0 A C = Fℓ  M21 0 B D M11 I  0 0 = Fℓ  M21 0 M11 I for any nonsingular matrix E. −1 −1 C = M12 , and D = −M22 M12 ;   −I −1  M22  , −M12 0   −I  , E −1  M22 M12 + E (b) If M21 is invertible, then Fℓ (M, Q) = (A + BQ)(C + DQ)−1 −1 −1 −1 −1 with A = M11 M21 , B = M12 − M11 M21 M22 , C = M21 , and D = −M21 M22 ; that is,      0 M12 M11 A B −1  I  , −M21 0 = Fℓ  0 C D −I M22 0    M11 0 M12  , E −1  I 0 = Fℓ  0 −I M22 M21 + E for any nonsingular matrix E. However, for an arbitrary LFT Fℓ (M, Q), neither M21 nor M12 is necessarily square and invertible; therefore, the alternative fractional formula is more restrictive. It should be pointed out that some seemingly simple functions do not have simple LFT representations. For example, (A + QB)(I + QD)−1 cannot always be written in the form of Fℓ (M, Q) for some M ; however, it can be written as (A + QB)(I + QD)−1 = Fℓ (N, ∆) 9.1. Linear Fractional Transformations with  A N =  −B D 169  A −B  , D I 0 0 ∆=  Q Q  . Note that the dimension of ∆ is twice of Q. The following lemma shows that the inverse of an LF T is still an LFT.   M11 M12 and M22 is nonsingular. Then Lemma 9.3 Let M = M21 M22 (Fu (M, ∆))−1 = Fu (N, ∆) with N , is given by N=  −1 M11 − M12 M22 M21 −1 M22 M21 −1 −M12 M22 −1 M22  . LFT is a very convenient tool to formulate many mathematical objects. We shall illustrate this by the following two examples. Simple Block Diagrams A feedback system with the following block diagram ✛ W1 ✛ uf i y ✲ K −✻ u d ❄ ✲ i ✲ P ✲ W2 v✲ ❄ n i✛ F ✛ can be rearranged as an LFT: ✛z y ✛w G ✛ ✲ K with w=  d n  , z=  v uf  u  W2 P , G= 0 −F P 0 0 −F  W2 P W1  . −F P 170 LINEAR FRACTIONAL TRANSFORMATION A state-space realization for the generalized plant G can be obtained by directly realizing the transfer matrix G using any standard multivariable realization techniques (e.g., Gilbert realization). However, the direct realization approach is usually complicated. Here we shall show another way to obtain the realization for G based on the realizations of each component. To simplify the expression, we shall assume that the plant P is strictly proper and P , F , W1 , and W2 have, respectively, the following state-space realizations:         Af Bf Au Bu Av Bv Ap Bp , W1 = , W2 = . , F = P = Cp 0 Cf Df Cu Du Cv Dv That is, ẋp = Ap xp + Bp (d + u), yp = Cp xp , ẋf = Af xf + Bf (yp + n), −y = Cf xf + Df (yp + n), ẋu = Au xu + Bu u, uf = Cu xu + Du u, ẋv = Av xv + Bv yp , v = Cv xv + Dv yp . Now define a new state vector  xp  xf   x=  xu  xv  and eliminate the variable yp to get a realization of G as ẋ = Ax + B1 w + B2 u z = C1 x + D11 w + D12 u y = C2 x + D21 w + D22 u with  Ap 0 0  Bf Cp Af 0  A= 0 0 Au Bv Cp 0 0  Dv Cp 0 C1 = 0 0  C2 = −Df Cp −Cf     Bp 0 0 Bp    0   , B1 =  0 Bf  , B2 =  0  0  Bu 0  0  Av 0 0 0    0 0 Cv , D11 = 0, D12 = Cu 0 Du    0 0 , D21 = 0 −Df , D22 = 0.     9.1. Linear Fractional Transformations 171 Parametric Uncertainty: A Mass/Spring/Damper System One natural type of uncertainty is unknown coefficients in a state-space model. To motivate this type of uncertainty description, we shall begin with a familiar mechanical system, shown in Figure 9.1. ✻ F m ❳ ❳ ✘ ✘ k ✘ ❳❳❳ ✘ ✘ ✘ ❳ ❳ c Figure 9.1: A mass/spring/damper system The dynamical equation of the system motion can be described by ẍ + c k F ẋ + x = . m m m Suppose that the three physical parameters m, c, and k are not known exactly, but are believed to lie in known intervals. In particular, the actual mass m is within 10% of a nominal mass, m̄, the actual damping value c is within 20% of a nominal value of c̄, and the spring stiffness is within 30% of its nominal value of k̄. Now introducing perturbations δm , δc , and δk , which are assumed to be unknown but lie in the interval [−1, 1], the block diagram for the dynamical system is as shown in Figure 9.2. It is easy to check that 1 m can be represented as an LFT in δm : 1 1 1 0.1 = = − δm (1 + 0.1δm )−1 = Fℓ (M1 , δm ) m m̄(1 + 0.1δm ) m̄ m̄ # − 0.1 m̄ . Suppose that the input signals of the dynamical system with M1 = 1 −0.1 are selected as x1 = x, x2 = ẋ, F , and the output signals are selected as ẋ1 and ẋ2 . To represent the system model as an LFT of the natural uncertainty parameters δm , δc , and δk , we shall first isolate the uncertainty parameters and denote the inputs and outputs of δk , δc , and δm as yk , yc , ym and uk , uc , um , respectively, as shown in Figure 9.3. " 1 m̄ 172 LINEAR FRACTIONAL TRANSFORMATION x 1 s ẋ ✛ ✛ẍ 1 s ✛ 1 m̄(1+0.1δm ) ✲ c̄(1 + 0.2δc ) d✛ ✻ − F ✲d ✻ + ✲ k̄(1 + 0.3δk ) Figure 9.2: Block diagram of mass/spring/damper equation x1 1 s ✛ x2 1 s ✛ ym M1 ✲ δm ✲ c̄ ✲ 0.2 ✲ δc yc ✲ k̄ ✲ 0.3 ✲ δ yk k e✛ F ✻− ✛ ✛ um ✲e ✲e ✻ ✻ uc ✲e ✻ uk Figure 9.3: A block diagram for the mass/spring/damper system with uncertain parameters 9.2. Basic Principle 173 Then  ẋ1   ẋ2   yk    yc ym   0   k̄   − m̄    =  0.3k̄       0 −k̄ 1 c̄ − m̄ 0 0.2c̄ −c̄ 0 0 1 − m̄ 0 0 −1 1 m̄ 0 0 1 That is, " ẋ1 ẋ2 # where  0  k̄  − m̄  M =  0.3k̄   0 −k̄ 9.2 1 0 c̄ − m̄ 0 0.2c̄ −c̄ 1 m̄ 0 0 1 0 1 − m̄ 0 0 −1 0 1 − m̄ 0 0 −1 0 − 0.1 m̄ 0 0 −0.1             x1 x2 F uk uc um         uk yk        ,  u c  = ∆  yc  .   um ym    x1   = Fℓ (M, ∆)  x2  F  0 1 − m̄ 0 0 −1 0 − 0.1 m̄ 0 0 −0.1     δk  , ∆ =   0   0  0 δc 0  0  0 . δm Basic Principle We have studied two simple examples of the use of LFTs and, in particular, their role in modeling uncertainty. The basic principle at work here in writing a matrix LFT is often referred to as “pulling out the ∆’s”. We will try to illustrate this with another picture. Consider a structure with four substructures interconnected in some known way, as shown in Figure 9.4. This diagram can be redrawn as a standard one via “pulling out the ∆’s” in Figure 9.5. Now the matrix M of the LFT can be obtained by computing the corresponding transfer matrix in the shadowed box. We shall illustrate the preceding principle with an example. Consider an input/output relation a + bδ2 + cδ1 δ22 z= w =: Gw 1 + dδ1 δ2 + eδ12 where a, b, c, d, and e are given constants or transfer functions. We would like to write G as an LFT in terms of δ1 and δ2 . We shall do this in three steps: 1. Draw a block diagram for the input/output relation with each δ separated as shown in Figure 9.6. 174 LINEAR FRACTIONAL TRANSFORMATION .. .. . . . . . .. .. .. .. .. .. .. .. .. . . .. .. . . ..... . . . . . ............................. . . ....... .. .. . . .... ∆1 ✛ ......................... .. ... .... .. .. .. . . . .. . .. . . .. ............... ... . . . . . .. ✛ . . ............... ........✛ ∆2 ✛ ... .. . . . . .. . . . . ...... .... . . ............... ...... .... . .. . . .... .. ... . . ......... .. .. ... .. .. .. . . .. ... .. .. . . ............ ................... ...................✛ . . . . . . . . . . . . . . . . . .. . ................... .... . . . . ...... .... ✛ .. . . . . ∆3 .. .. .. .. .. . .. . . . .. .... ✛ K ✛ ...... . . . . .. . . . . .. .. . .... . . .. . ... .................. .. ✛ . .. .. .. ......... ........................ ....... . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . .. .. ✛ Figure 9.4: Multiple source of uncertain structure ✲ ∆1 ✲ ∆2 ✛ ✛ ✛ ✛ ✛ ✲ ∆3 .. .. . . . . . .. .. .. .. .. .. .. .. .. . . .. .. . . ..... . . . . . ............................. . . ....... .. ..✛ . ✛ ......................... .. ... .... .. .. .. .✛ . .... . . . . .✛ .. . . . .. ............... ... . . ... . . . . . ✛ . . ............... ........✛ ❄ . ✛.... . .. . . . . ...... .... . . ............... ...... .... . . . . . . . . . . . . . . . . .. . .. . .. .. .. . . .. . . . .. .. .. ... .. .. . . ............ ................... ...................✛ . . . . . . . .. . .. . .. .. .. .. .. .. .. .. .. .. .. .... .. . ✛........ ......... ..... .. . . . . . .. ............ .. .. ✛ ....... .... ✛ . . . . .... . . .. . ... .................. .. ✛ .......✛ .. ......... ........................ ....... . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . .. .. ✲ K Figure 9.5: Pulling out the ∆’s 9.2. Basic Principle 175 a ✛ z ✛ u ❄ e✛ 4 y4 δ2 ✛ e✛ ✻ b ✛ c ✻ u3 δ2 ✛ y3 ❄ −d u1 δ1 ✛ y2 ✲ δ1 y1 e✛ ✻ u2 ✲ −e w ✲e ✻ Figure 9.6: Block diagram for G 2. Mark the inputs and outputs of the δ’s as y’s and u’s, respectively. (This is essentially pulling out the ∆’s.) 3. Write z and y’s in terms of w and u’s with all δ’s taken out. (This step is equivalent to computing the transformation in the shadowed box in Figure 9.5.)   y1     y2       y3  = M         y4   z  where Then     M =    0 1 1 0 0  u1  u2   u3    u4  w −e −d 0 0 0 0 −be −bd + c −ae −ad z = Fu (M, ∆)w, ∆ = " 0 0 0 0 1 δ1 I2 0 1 0 0 b a     .    0 δ2 I2 # . All LFT examples in Section 9.1 can be obtained following the preceding steps. 176 LINEAR FRACTIONAL TRANSFORMATION For Simulink users, it is much easier to do all the computations using Simulink block diagrams, as shown in the following example. Example 9.1 Consider the HIMAT (highly maneuverable aircraft) control problem from the µ Analysis and Synthesis Toolbox (Balas et al. [1994]). The system diagram is shown in Figure 9.7 where     50(s + 100) 0.5(s + 3) 0 0     Wdel =  s + 10000 , Wp =  s + 0.03 , 50(s + 100)  0.5(s + 3)  0 0 s + 10000 s + 0.03   2(s + 1.28) 0   Wn =  s + 320 2(s + 1.28)  , 0 s + 320   0 0 −0.0226 −36.6 −18.9 −32.1    −0.414 0  0 −1.9 0.983 0    0.0123 −11.7 −2.63 0 −77.8 22.4    P0 =     0 0 1 0 0 0    0 57.3 0 0 0 0    0 0 0 57.3 0 0 " z1 z2 # " ✲ ∆ p1 p2 " # d1 d2 # Wdel ✻ ❄✲ ✲f P0 u K ✛ ❄ ✲f y ✲ Wp ❄ f✛ Wn ✛ Figure 9.7: HIMAT closed-loop interconnection ✲ " e1 e2 # " n1 n2 # 9.2. Basic Principle 177 The open-loop interconnection is            z1 z2 e1 e2 y1 y2   p1    p2      d   1        d 2    .  = Ĝ(s)    n 1           n2   u   1  u2  The Simulink block diagram of this open-loop interconnection is shown in Figure 9.8. Figure 9.8: Simulink block diagram for HIMAT (aircraft.m) The Ĝ(s) = " A B C D # can be computed by ≫ [A, B, C, D] = linmod(′ aircraft′ ) 178 LINEAR FRACTIONAL TRANSFORMATION which gives  −10000I2 0 0 0 0 0 0 0 0 −0.0226 −36.6 −18.9 −32.1 0 0 0       A=       0 0 −1.9 0.983 0 0 0 0 0 0.0123 −11.7 −2.63 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −54.087 0 0 −0.018 0 0 0 0 0 0 −54.087 0 −0.018 0 0 0 0 0 0 0 0 −320I2  0  0    0   −0.4140 B=  −77.8   0    0 0     C=   0 0 0 −703.5624 0 0 0 0 0 0 0 0  0 0 0 0 −0.4140 0 0 −77.8 0 22.4 0 0 0 0 0 0 −0.9439I2 0 0 0 0 0 −25.2476I2 0 0            703.5624I2 0 0 0 0 0 0 0 0 0 0 28.65 0 0 −0.9439 0 0 0 0 0 0 0 28.65 0 −0.9439 0 0 0 0 57.3 0 0 0 0 25.2476 0 0 0 0 0 57.3 0 0 0 25.2476   0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 0 0 0.5 0 0 0 0 0 0 0 0.5 0 0 0 50   0 0 1 0 2 0 0 0 0 0 1 0 2 0              −703.5624   0 22.4     D=    9.3          0  . 0   0  0 Redheffer Star Products The most important property of LFTs is that any interconnection of LFTs is again an LFT. This property is by far the most often used and is the heart of LFT machinery. Indeed, it is not hard to see that most of the interconnection structures discussed earlier (e.g., feedback and cascade) can be viewed as special cases of the so-called star product. 9.3. Redheffer Star Products 179 Suppose that P and K are compatibly partitioned matrices " # " # P11 P12 K11 K12 P = , K= P21 P22 K21 K22 such that the matrix product P22 K11 is well-defined and square, and assume further that I − P22 K11 is invertible. Then the star product of P and K with respect to this partition is defined as # " −1 P12 (I − K11 P22 ) K12 Fl (P, K11 ) . (9.1) P ⋆ K := −1 K21 (I − P22 K11 ) P21 Fu (K, P22 ) Note that this definition is dependent on the partitioning of the matrices P and K. In fact, this star product may be well-defined for one partition and not well-defined for another; however, we will not explicitly show this dependence because it is always clear from the context. In a block diagram, this dependence appears, as shown in Figure 9.9. z ✛ P ẑ ✛ w ✛ ✛ y❛ u ❛❛✦✦✦ ✦ ❛ ❛❛ ✦✦ ✛ K ✛ z ✛ ẑ ✛ P ⋆K ✛w ✛ ŵ ŵ Figure 9.9: Interconnection of LFTs Now suppose that P and  A B1  P =  C1 D11 C2 D21 Then the transfer matrix K are transfer matrices with   AK B2   K =  CK1 D12  D22 CK2 P ⋆K : "  B̄1 w ŵ # 7→ " z ẑ state-space representations:  BK1 BK2  DK11 DK12  . DK21 # has a representation Ā  P ⋆K =  C̄1 C̄2 D̄11 D̄21 B̄2 D̄12 D̄22   =  " Ā B̄ C̄ D̄ # DK22 180 LINEAR FRACTIONAL TRANSFORMATION where # Ā = " A + B2 R̃−1 DK11 C2 BK1 R−1 C2 B̄ = " B1 + B2 R̃−1 DK11 D21 BK1 R−1 D21 B2 R̃−1 DK12 BK2 + BK1 R−1 D22 DK12 # = " C1 + D12 DK11 R−1 C2 D12 R̃−1 CK1 DK21 R−1 C2 CK2 + DK21 R−1 D22 CK1 # = " D11 + D12 DK11 R−1 D21 D12 R̃−1 DK12 DK21 R−1 D21 DK22 + DK21 R−1 D22 DK12 C̄ D̄ B2 R̃−1 CK1 AK + BK1 R−1 D22 CK1 R = I − D22 DK11 , In fact, it is easy to show that " A Ā = C2 " B1 B̄ = D21 " C1 C̄ = C2 " D11 D̄ = D21 B2 D22 B2 D22 D12 D22 D12 D22 # R̃ = I − DK11 D22 . # " ⋆ CK1 BK1 AK DK11 BK1 DK12 BK2 # CK1 # " ⋆ # " ⋆ # DK11 DK11 , # , , DK21 CK2 # " # DK11 DK12 ⋆ . DK21 DK22 The Matlab command starp can be used to compute the star product: ≫ P ⋆ K = starp(P, K, dimy, dimu) where dimy and dimu are the dimensions of y and u, respectively. In the particular case when dim(ẑ) = 0 and dim(ŵ) = 0, we have ≫ Fℓ (P, K) = starp(P, K) 9.4 Notes and References This chapter is based on the lecture notes by Packard [1991] and the paper by Doyle, Packard, and Zhou [1991]. 9.5. Problems 9.5 181 Problems ˜ = M ∆N , where ∆ is block diagoProblem 9.1 Find M and N matrices such that ∆ nal. i h ˜ = ∆1 ∆2 1. ∆ ˜ = 2. ∆   ∆1 0 0 0 0 ∆2  h 0 ∆1 " # ˜ = 3. ∆  ∆2 0  ∆1 ˜ =  ∆2 4. ∆ 0 0 ∆3 0  ∆1 ˜ =  ∆2 5. ∆ 0 ∆3 ∆4 ∆5 0 ∆3  0 0  ∆4 i     0 ∆6  0 −1 Problem 9.2 Let G = (I − P (s)∆) . Find a matrix M (s) such that G = Fu (M, ∆). Problem 9.3 Consider the unity feedback system with G(s) of size 2 × 2. Suppose G(s) has an uncertainty model of the form G(s) = " [1 + ∆11 (s)]g11 (s) [1 + ∆21 (s)]g21 (s) [1 + ∆12 (s)]g12 (s) [1 + ∆22 (s)]g22 (s) # . Suppose also that we wish to study robust stability of the feedback system. Pull out the ∆’s and draw the appropriate block diagram in the form of a structured perturbation of a nominal system. Problem 9.4 Let  A  P (s) =  C1 C2 B1 D11 D21 B2 D12 D22   , K(s) = Find state-space realizations for Fℓ (P, K) and Fℓ (P, D̂). " Â B̂ Ĉ D̂ # . 182 LINEAR FRACTIONAL TRANSFORMATION Problem 9.5 Suppose D21 is nonsingular and  A B1  M (s) =  C1 D11 C2 D21 Find a state-space realization for  " B2   D12  . D22 # # " M11 0 M12  I 0 M̂(s) =  i  h 0 M21 + E −I M22     where E is a constant matrix. Find the state-space realization for Fℓ (M̂ , E −1 ) when E = I. Chapter 10 µ and µ Synthesis It is noted that the robust stability and robust performance criteria derived in Chapter 8 vary with the assumptions about the uncertainty descriptions and performance requirements. We shall show in this chapter that they can all be treated in a unified framework using the LFT machinery introduced in the last chapter and the structured singular value to be introduced in this chapter. This, of course, does not mean that those special problems and their corresponding results are not important; on the contrary, they are sometimes very enlightening to our understanding of complex problems, such as those in which complex problems are formed from simple problems. On the other hand, a unified approach may relieve the mathematical burden of dealing with specific problems repeatedly. Furthermore, the unified framework introduced here will enable us to treat exactly the robust stability and robust performance problems for systems with multiple sources of uncertainties, which is a formidable problem from the standpoint of Chapter 8, in the same fashion as single unstructured uncertainty. Indeed, if a system is subject to multiple sources of uncertainties, in order to use the results in Chapter 8 for unstructured cases, it is necessary to reflect all sources of uncertainties from their known point of occurrence to a single reference location in the loop. Such reflected uncertainties invariably have a great deal of structure, which must then be “covered up” with a large, arbitrarily more conservative perturbation in order to maintain a simple cone-bounded representation at the reference location. Readers might have some idea about the conservativeness in such reflection based on the skewed specification problem, where an input multiplicative uncertainty of the plant is reflected at the output and the size of the reflected uncertainty is proportional to the condition number of the plant. In general, the reflected uncertainty may be proportional to the condition number of the transfer matrix between its original location and the reflected location. Thus it is highly desirable to treat the uncertainties as they are and where they are. The structured singular value is defined exactly for that purpose. 183 184 10.1 µ AND µ SYNTHESIS General Framework for System Robustness As we illustrated in Chapter 9, any interconnected system may be rearranged to fit the general framework in Figure 10.1. Although the interconnection structure can become quite complicated for complex systems, many software packages, such as Simulink and µ Analysis and Synthesis Toolbox, are available that could be used to generate the interconnection structure from system components. Various modeling assumptions will be considered, and the impact of these assumptions on analysis and synthesis methods will be explored in this general framework. ✲ ∆ z✛ P ✛ ✛ ✛ w ✲ K Figure 10.1: General framework Note that uncertainty may be modeled in two ways, either as external inputs or as perturbations to the nominal model. The performance of a system is measured in terms of the behavior of the outputs or errors. The assumptions that characterize the uncertainty, performance, and nominal models determine the analysis techniques that must be used. The models are assumed to be FDLTI systems. The uncertain inputs are assumed to be either filtered white noise or weighted power or weighted Lp signals. Performance is measured as weighted output variances, or as power, or as weighted output Lp norms. The perturbations are assumed to be themselves FDLTI systems that are norm-bounded as input-output operators. Various combinations of these assumptions form the basis for all the standard linear system analysis tools. Given that the nominal model is an FDLTI system, the interconnection system has the form   P11 (s) P12 (s) P13 (s)   P (s) =  P21 (s) P22 (s) P23 (s)  P31 (s) P32 (s) P33 (s) and the closed-loop system is an LFT on the perturbation and the controller given by z = Fu (Fℓ (P, K), ∆) w = Fℓ (Fu (P, ∆), K) w. We shall focus our discussion in this section on analysis methods; therefore, the controller may be viewed as just another system component and absorbed into the 10.1. General Framework for System Robustness 185 interconnection structure. Denote M (s) = Fℓ (P (s), K(s)) = " M11 (s) M21 (s) M12 (s) M22 (s) # . Then the general framework reduces to Figure 10.2, where   z = Fu (M, ∆)w = M22 + M21 ∆(I − M11 ∆)−1 M12 w. ✲ ∆ z ✛ M ✛ ✛ w Figure 10.2: Analysis framework Suppose K(s) is a stabilizing controller for the nominal plant P . Then M (s) ∈ RH∞ . In general, the stability of Fu (M, ∆) does not necessarily imply the internal stability of the closed-loop feedback system. However, they can be made equivalent with suitably chosen w and z. For example, consider again the multiplicatively perturbed system shown in Figure 10.3. ✲ W2 d2 e1 e2 d1 ✲e ✲ K ✲❄ e ✲ P −✻ e3 d3 ✲ ∆ ✲ W1 ✲❄ e Figure 10.3: Multiplicatively perturbed systems Now let w := " d1 d2 # , z := " e1 e2 # . Then the system is robustly stable for all ∆(s) ∈ RH∞ with k∆k∞ < 1 if and only if Fu (M, ∆) ∈ RH∞ for all admissible ∆ with M11 = −W2 P K(I + P K)−1 W1 , which is guaranteed by kM11 k∞ ≤ 1. The analysis results presented in the previous chapters together with the associated synthesis tools are summarized in Table 10.1 with various uncertainty modeling assumptions. However, the analysis is not so simple for systems with multiple sources of model uncertainties, including the robust performance problem for systems with unstructured 186 µ AND µ SYNTHESIS Input Assumptions Performance Specifications E(w(t)w(τ )∗ ) = δ(t − τ )I E(z(t)∗ z(t)) ≤ 1 Perturbation Assumptions E(U0 U0∗ ) = I 2 E(kzk2 ) Synthesis Methods LQG ∆=0 w = U0 δ(t) Analysis Tests kM22 k2 ≤ 1 Wiener-Hopf ≤1 H2 kwk2 ≤ 1 kzk2 ≤ 1 ∆=0 kM22 k∞ ≤ 1 Singular Value Loop Shaping kwk2 ≤ 1 Internal Stability k∆k∞ < 1 kM11 k∞ ≤ 1 H∞ Table 10.1: General analysis for single source of uncertainty uncertainty. As we shown in Chapter 9, if a system is built from components that are themselves uncertain, then, in general, the uncertainty in the system level is structured, involving typically a large number of real parameters. The stability analysis involving real parameters is much more difficult and will be discussed in Chapter 18. Here we shall simply cover the real parametric uncertainty with norm-bounded dynamical uncertainty. Moreover, the interconnection model M can always be chosen so that ∆(s) is block diagonal, and, by absorbing any weights, k∆k∞ < 1. Thus we shall assume that ∆(s) takes the form of ∆(s) = {diag [δ1 Ir1 , . . . , δs IrS , ∆1 , . . . , ∆F ] : δi (s) ∈ RH∞ , ∆j ∈ RH∞ } with kδi k∞ < 1 and k∆j k∞ < 1. Then the system is robustly stable iff the interconnected system in Figure 10.4 is stable. The results of Table 10.1 can be applied to analysis of the system’s robust stability in two ways: (1) kM11 k∞ ≤ 1 implies stability, but not conversely, because this test ignores the known block diagonal structure of the uncertainties and is equivalent to regarding ∆ as unstructured. This can be arbitrarily conservative in that stable systems can have arbitrarily large kM11 k∞ . 10.2. Structured Singular Value 187 ✲ ∆F (s) .. . ✲ δ1 (s)I ✛ .. . M11 (s) ✛ .. . Figure 10.4: Robust stability analysis framework (2) Test for each δi (∆j ) individually (assuming no uncertainty in other channels). This test can be arbitrarily optimistic because it ignores interaction between the δi (∆j ). This optimism is also clearly shown in the spinning body example in Section 8.6. The difference between the stability margins (or bounds on ∆) obtained in (1) and (2) can be arbitrarily far apart. Only when the margins are close can conclusions be made about the general case with structured uncertainty. The exact stability and performance analysis for systems with structured uncertainty requires a new matrix function called the structured singular value (SSV), which is denoted by µ. 10.2 Structured Singular Value 10.2.1 Definitions of µ We shall motivate the definition of the structured singular value by asking the following question: Given a matrix M ∈ Cp×q , what is the smallest perturbation matrix ∆ ∈ Cq×p in the sense of σ(∆) such that det(I − M ∆) = 0? That is, we are interested in finding  αmin := inf σ(∆) : det(I − M ∆) = 0, ∆ ∈ Cq×p . 188 µ AND µ SYNTHESIS It is easy to see that  αmin = inf α : det(I − αM ∆) = 0, σ(∆) ≤ 1, ∆ ∈ Cq×p = 1 max ρ(M ∆) σ(∆)≤1 and max ρ(M ∆) = σ(M ). σ(∆)≤1 Hence the smallest norm of a “destabilizing” perturbation matrix is 1/σ(M ) with a smallest “destabilizing” ∆: ∆des = 1 v1 u∗1 , σ(M ) det(I − M ∆des ) = 0 where M = σ(M )u1 v1∗ + σ2 u2 v2∗ + · · · is a singular value decomposition. So the reciprocal of the largest singular value of a matrix is a measure of the smallest “destabilizing” perturbation matrix. Hence it is instructive to introduce the following alternative definition for the largest singular value: σ(M ) := 1 . inf{σ(∆) : det(I − M ∆) = 0, ∆ ∈ Cq×p } Next we consider a similar problem but with ∆ structurally restricted. In particular, we consider the block diagonal matrix ∆. We shall consider two types of blocks: repeated scalar and full blocks. Let S and F represent the number of repeated scalar blocks and the number of full blocks, respectively. To bookkeep their dimensions, we introduce positive integers r1 , . . . , rS ; m1 , . . . , mF . The ith repeated scalar block is ri × ri , while the jth full block is mj × mj . With those integers given, we define ∆ ⊂ Cn×n as  (10.1) ∆ = diag [δ1 Ir1 , . . . , δs IrS , ∆1 , . . . , ∆F ] : δi ∈ C, ∆j ∈ Cmj ×mj . For consistency among all the dimensions, we must have S X i=1 ri + F X mj = n. j=1 Often, we will need norm-bounded subsets of ∆, and we introduce the following notation: B∆ = {∆ ∈ ∆ : σ(∆) ≤ 1} (10.2) Bo ∆ = {∆ ∈ ∆ : σ(∆) < 1} (10.3) where the superscript “o” symbolizes the open ball. To keep the notation as simple as possible in equation (10.1), we place all of the repeated scalar blocks first; in actuality, they can come in any order. Also, the full blocks do not have to be square, but restricting them as such saves a great deal in terms of notation. 10.2. Structured Singular Value 189 Now we ask a similar question: Given a matrix M ∈ Cp×q , what is the smallest perturbation matrix ∆ ∈ ∆ in the sense of σ(∆) such that det(I − M ∆) = 0? That is, we are interested in finding αmin := inf {σ(∆) : det(I − M ∆) = 0, ∆ ∈ ∆} . Again we have αmin = inf {α : det(I − αM ∆) = 0, ∆ ∈ B∆} = 1 . max ρ(M ∆) ∆∈B∆ Similar to the unstructured case, we shall call 1/αmin the structured singular value and denote it by µ∆ (M ). Definition 10.1 For M ∈ Cn×n , µ∆ (M ) is defined as µ∆ (M ) := 1 min {σ(∆) : ∆ ∈ ∆, det (I − M ∆) = 0} unless no ∆ ∈ ∆ makes I − M ∆ singular, in which case (10.4) µ∆ (M ) := 0. Remark 10.1 Without a loss in generality, the full blocks in the minimal norm ∆ can each be chosen to be dyads (rank = 1). To see this, assume S = 0 (i.e., all blocks are full blocks). Suppose that I − M ∆ is singular for some ∆ ∈ ∆. Then there is an x ∈ Cn such that M ∆x = x. Now partition x conformably with ∆:   x1    x2  mi  x=  ..  , xi ∈ C , i = 1, . . . , F  .  xF and let Define  ∆i xi x∗i     kx k2 , xi 6= 0; i ˜i = ∆     0, xi = 0 for i = 1, 2, . . . , F. ˜ = diag{∆ ˜ 1, ∆ ˜ 2, . . . , ∆ ˜ F }. ∆ ˜ ≤ σ(∆), ∆x ˜ = ∆x, and thus (I − M ∆)x ˜ = (I − M ∆)x = 0 (i.e., I − M ∆ ˜ Then σ(∆) is also singular). Hence we have replaced a general perturbation ∆ that satisfies the ˜ that is no larger [in the σ(·) sense] and has singularity condition with a perturbation ∆ rank 1 for each block but still satisfies the singularity condition. ✸ 190 µ AND µ SYNTHESIS Lemma 10.1 µ∆ (M ) = max ρ (M ∆) ∆∈B∆ In view of this lemma, continuity of the function µ : Cn×n → R is apparent. In general, though, the function µ : Cn×n → R is not a norm, since it does not satisfy the triangle inequality; however, for any α ∈ C, µ (αM ) = |α|µ (M ), so in some sense, it is related to how “big” the matrix is. We can relate extreme sets. µ∆ (M ) to familiar linear algebra quantities when ∆ is one of two • If ∆ = {δI : δ ∈ C} (S = 1, F = 0, r1 = n), then radius of M . µ∆ (M ) = ρ (M ), the spectral Proof. The only ∆’s in ∆ that satisfy the det (I − M ∆) = 0 constraint are reciprocals of nonzero eigenvalues of M . The smallest one of these is associated with the largest (magnitude) eigenvalue, so, µ∆ (M ) = ρ (M ). ✷ • If ∆ = Cn×n (S = 0, F = 1, m1 = n), then µ∆ (M ) = σ(M ). Obviously, for a general ∆, as in equation (10.1), we must have {δIn : δ ∈ C} ⊂ ∆ ⊂ Cn×n . (10.5) Hence directly from the definition of µ and from the two preceding special cases, we conclude that ρ (M ) ≤ µ∆ (M ) ≤ σ (M ) . (10.6) These bounds alone are not sufficient for our purposes because the gap between ρ and σ can be arbitrarily large. Example 10.1 Suppose ∆= " δ1 0 0 δ2 # and consider " # 0 β (1) M = for any β > 0. Then ρ(M ) = 0 and σ(M ) = β. But µ(M ) = 0 0 0 since det(I − M ∆) = 1 for all admissible ∆. " # −1/2 1/2 (2) M = . Then ρ(M ) = 0 and σ(M ) = 1. Since −1/2 1/2 det(I − M ∆) = 1 +  it is easy to see that min maxi |δi | : 1 + δ1 −δ2 2 δ1 − δ2 , 2 = 0 = 1, so µ(M ) = 1. 10.2. Structured Singular Value 191 Thus neither ρ nor σ provide useful bounds even in simple cases. The only time they do provide reliable bounds is when ρ ≈ σ. However, the bounds can be refined by considering transformations on M that do not affect µ∆ (M ), but do affect ρ and σ. To do this, define the following two subsets of Cn×n : U = {U ∈ ∆ : U U ∗ = In } D = ( (10.7)   ) diag D1 , . . . , DS , d1 Im1 , . . . , dF −1 ImF −1 , ImF : . Di ∈ Cri ×ri , Di = Di∗ > 0, dj ∈ R, dj > 0 (10.8) Note that for any ∆ ∈ ∆, U ∈ U, and D ∈ D, U∗ ∈ U U∆ ∈ ∆ ∆U ∈ ∆ σ (U ∆) = σ (∆U ) = σ (∆) D∆ = ∆D. (10.9) (10.10) Consequently, we have the following: Theorem 10.2 For all U ∈ U and D ∈ D
µ∆ (M U ) = µ∆ (U M ) = µ∆ (M ) = µ∆ DM D−1 . Proof. For all D ∈ D and ∆ ∈ ∆, (10.11)

det (I − M ∆) = det I − M D−1 ∆D = det I − DM D−1 ∆
since D commutes with ∆. Therefore µ∆ (M ) = µ∆ DM D−1 . Also, for each U ∈ U, det (I − M ∆) = 0 if and only if det (I − M U U ∗ ∆) = 0. Since U ∗ ∆ ∈ ∆ and σ (U ∗ ∆) = σ (∆), we get µ∆ (M U ) = µ∆ (M ) as desired. The argument for U M is the same. ✷ Therefore, the bounds in equation (10.6) can be tightened to max ρ(U M ) ≤ max ρ (∆M ) = µ∆ (M ) ≤ inf σ DM D−1 U∈U ∆∈B∆ D∈D
(10.12) where the equality comes from Lemma 10.1. Note that the last element in the D matrix −1 is normalized to 1 since for any nonzero scalar γ, DM D−1 = (γD) M (γD) . Remark 10.2 Note that the scaling set D in Theorem 10.2 and in inequality (10.12) is not necessarily restricted to being Hermitian. In fact, it can be replaced by any set of nonsingular matrices that satisfy equation (10.10). However, enlarging the set of scaling matrices does not improve the upper-bound in inequality (10.12). This can be shown 192 µ AND µ SYNTHESIS as follows: Let D be any nonsingular matrix such that D∆ = ∆D. Then there exist a Hermitian matrix 0 < R = R∗ ∈ D and a unitary matrix U such that D = U R and


inf σ DM D−1 = inf σ U RM R−1 U ∗ = inf σ RM R−1 . D D R∈D Therefore, there is no loss of generality in assuming D to be Hermitian. 10.2.2 ✸ Bounds In this section we will concentrate on the bounds
max ρ (U M ) ≤ µ∆ (M ) ≤ inf σ DM D−1 . U∈U D∈D The lower bound is always an equality (Doyle [1982]). Theorem 10.3 max ρ(M U ) = µ∆ (M ) U∈U Unfortunately, the quantity ρ (U M ) can have multiple local maxima that are not global. Thus local search cannot be guaranteed to obtain µ, but can only yield a lower bound. For computation purposes one can derive a slightly different formulation of the lower bound as a power algorithm that is reminiscent of power algorithms for eigenvalues and singular values (Packard and Doyle [1988a, 1988b]). While there are open questions about convergence, the algorithm usually works quite well and has proven to be an effective method to compute µ. The upper-bound can be reformulated as a convex optimization problem, so the global minimum can, in principle, be found. Unfortunately, the upper-bound is not always equal to µ. For block structures ∆ satisfying 2S + F ≤ 3, the upper-bound is always equal to µ∆ (M ), and for block structures with 2S + F > 3, there exist matrices for which µ is less than the infimum. This can be summarized in the following diagram, which shows for which cases the upper-bound is guaranteed to be equal to µ. See Packard and Doyle [1993] for details. Theorem 10.4 µ∆ (M ) = inf σ(DM D−1 ) if 2S + F ≤ 3 D∈D F= 0 1 2 3 4 yes yes yes no S= 0 1 yes yes no no no 2 no no no no no Several of the boxes have connections with standard results. • S = 0, F = 1 : µ∆ (M ) = σ (M ). 10.2. Structured Singular Value • S = 1, F = 0 : 193
µ∆ (M ) = ρ (M ) = inf σ DM D−1 . This is a standard result in D∈D linear algebra. In fact, without a loss in generality, the matrix M can be assumed in Jordan canonical form. Now let     λ 1 1         λ 1 k         . . . .. .. .. J1 =   , D1 =   ∈ Cn1 ×n1 .         λ 1  k n1 −2    λ k n1 −1 Then inf D1 ∈Cn1 ×n1 σ(D1 J1 D1−1 ) = lim σ(D1 J1 D1−1 ) = |λ|. (Note that by Rek→∞ mark 10.2, the scaling matrix does not need to be Hermitian.) The conclusion follows by applying this result to each Jordan block. That µ equals to the preceding upper-bound in this case is also equivalent to the fact that Lyapunov asymptotic stability and exponential stability are equivalent for discrete time systems. This is because ρ (M ) < 1 (exponential stability of a discrete time system matrix M ) implies for some nonsingular D ∈ Cn×n σ(DM D−1 ) < 1 or (D−1 )∗ M ∗ D∗ DM D−1 − I < 0, which, in turn, is equivalent to the existence of a P = D∗ D > 0 such that M ∗P M − P < 0 (Lyapunov asymptotic stability). • S = 0, F = 2 : This case was studied by Redheffer [1959]. • S = 1, F = 1 : This is equivalent to a state-space characterization of the H∞ norm of a discrete time transfer function. • S = 2, F = 0 : This is equivalent to the fact that for multidimensional systems (two dimensional, in fact), exponential stability is not equivalent to Lyapunov stability. • S = 0, F ≥ 4 : For this case, the upper-bound is not always equal to µ. This is important, as these are the cases that arise most frequently in applications. Fortunately, the bound seems to be close to µ. The worst known example has a ratio of µ over the bound of about .85, and most systems are close to 1. The preceding bounds are much more than just computational schemes. They are also theoretically rich and can unify a number of apparently different results in linear systems theory. There are several connections with Lyapunov asymptotic stability, two of which were hinted at previously, but there are further connections between the 194 µ AND µ SYNTHESIS upper-bound scalings and solutions to Lyapunov and Riccati equations. Indeed, many major theorems in linear systems theory follow from the upper-bounds and from some results of linear fractional transformations. The lower bound can be viewed as a natural generalization of the maximum modulus theorem. Of course, one of the most important uses of the upper-bound is as a computational scheme when combined with the lower bound. For reliable use of the µ theory, it is essential to have upper and lower bounds. Another important feature of the upperbound is that it can be combined with H∞ controller synthesis methods to yield an ad hoc µ-synthesis method. Note that the upper-bound when applied to transfer functions is simply a scaled H∞ norm. This is exploited in the D − K iteration procedure to perform approximate µ synthesis (Doyle [1982]), which will be briefly introduced in Section 10.4. The upper and lower bounds of the structured singular value and the scaling matrix D can be computed using the MATLAB command ≫ [bounds,rowd] = mu(M,blk) where the structure of the ∆ is specified  δ1 I2 0   0 δ2   0 0  ∆=  0 0   0 0  0 0 by a two-column matrix blk. for example, a  0 0 0 0  0 0 0 0   ∆3 0 0 0    0 ∆4 0 0   0 0 δ5 I3 0   0 0 0 ∆6 δ1 , δ2 , δ5 , ∈ C, ∆3 ∈ C2×3 , ∆4 ∈ C3×3 , ∆6 ∈ C2×1 can be specified by       blk =      2 1 2 3 3 2 0 1 3 3 0 1       .     Note that ∆j is not required to be square. The outputs of the program include a 2 × 1 vector bounds containing the upper and lower bounds of µ∆ (M ) and the row vector rowd containing the scaling D. The D matrix can be recovered by ≫ [Dℓ , Dr ] = unwrapd(rowd, blk) where Dℓ and Dr denote the
left and right scaling matrices used in computing the upper-bound inf σ Dℓ M Dr−1 when some full blocks are not necessarily square and they are equal if all full blocks are square. 10.2. Structured Singular Value Example 10.2 Let  j   3+j   3+j   M =  −1 + j   3   1  2+j 2 2−j j −1 − j j 3 + 2j −1 − j 2j −1 + j 2 + 2j j 1+j 2 + 2j −1 195 0 2+j −1 + 2j 0 3j 3j 3 + 3j −1 −1 + j 3−j 1−j 1+j 1 + 2j 2 + 3j −1 + 3j 1 3j 2−j 3j 2+j 2j 2 + 3j −1 + j −1 + j 2 + 2j −j −1 + 2j 1−j and    δ1 I2       ∆= ∆=           Then blk =    and 2 1 2 2 0 1 3 1 ∆3     : δ1 , δ2 ∈ C, ∆3 ∈ C2×3 , ∆4 ∈ C2×1 .        ∆4  h i   and the Matlab program gives bounds = 10.5955 10.5518      Dℓ =     where                    δ2    Dr =    D1 = "  D1 0.7638 0.8809I3 1.0293       D1 0.7638 0.8809I2 1.0260 − 0.0657j −0.0701 + 0.3871j 1.0293I2      0.2174 − 0.3471j −0.4487 − 0.6953j # . In fact, Dℓ and Dr can be replaced by Hermitian matrices without changing the upperbound by replacing D1 with " # 1.0992 0.0041 − 0.0591j D̂1 = 0.0041 + 0.0591j 0.9215 196 µ AND µ SYNTHESIS since D1 = U1 D̂1 and U1 = " 0.9155 − 0.0713j −0.1029 + 0.3824j 0.2365 − 0.3177j −0.5111 − 0.7629j # is a unitary matrix. 10.2.3 Well-Posedness and Performance for Constant LFTs Let M be a complex matrix partitioned as " M11 M= M21 M12 M22 # (10.13) and suppose there are two defined block structures, ∆1 and ∆2 , which are compatible in size with M11 and M22 , respectively. Define a third structure ∆ as ) (" # ∆1 0 (10.14) ∆= : ∆1 ∈ ∆1 , ∆2 ∈ ∆2 . 0 ∆2 Now we may compute µ with respect to three structures. The notations we use to keep track of these computations are as follows: µ1 (·) is with respect to ∆1 , µ2 (·) is with respect to ∆2 , and µ∆ (·) is with respect to ∆. In view of these notations, µ1 (M11 ), µ2 (M22 ), and µ∆ (M ) all make sense, though, for instance, µ1 (M ) does not. This section is interested in following constant matrix problems: • Determine whether the LFT Fℓ (M, ∆2 ) is well-defined for all ∆2 ∈ ∆2 with σ(∆2 ) ≤ β (< β). • If so, determine how “large” Fℓ (M, ∆2 ) can get for this norm-bounded set of perturbations. Let ∆2 ∈ ∆2 . Recall that Fℓ (M, ∆2 ) is well-defined if I − M22 ∆2 is invertible. The first theorem is nothing more than a restatement of the definition of µ. Theorem 10.5 The linear fractional transformation Fℓ (M, ∆2 ) is well-defined (a) for all ∆2 ∈ B∆2 if and only if µ2 (M22 ) < 1. (b) for all ∆2 ∈ Bo ∆2 if and only if µ2 (M22 ) ≤ 1. As the “perturbation” ∆2 deviates from zero, the matrix Fℓ (M, ∆2 ) deviates from M11 . The range of values that µ1 (Fℓ (M, ∆2 )) takes on is intimately related to µ∆ (M ), as shown in the following theorem: 10.2. Structured Singular Value 197 Theorem 10.6 (main loop theorem) The following are equivalent:     µ2 (M22 ) < 1, and µ∆ (M ) < 1 ⇐⇒    max µ1 (Fℓ (M, ∆2 )) < 1 ∆2 ∈B∆2 µ∆ (M ) ≤ 1 ⇐⇒  µ2 (M22 ) ≤ 1, and       sup ∆2 ∈Bo ∆2 µ1 (Fℓ (M, ∆2 )) ≤ 1. Proof. We shall only prove the first part of the equivalence. The proof for the second part is similar. ⇐ Let ∆i ∈ ∆i be given, with σ (∆i ) ≤ 1, and define ∆ = diag [∆1 , ∆2 ]. Obviously ∆ ∈ ∆. Now " # I − M11 ∆1 −M12 ∆2 det (I − M ∆) = det . (10.15) −M21 ∆1 I − M22 ∆2 By hypothesis I − M22 ∆2 is invertible, and hence det (I − M ∆) becomes   −1 det (I − M22 ∆2 ) det I − M11 ∆1 − M12 ∆2 (I − M22 ∆2 ) M21 ∆1 . Collecting the ∆1 terms leaves det (I − M ∆) = det (I − M22 ∆2 ) det (I − Fℓ (M, ∆2 ) ∆1 ) . (10.16) But µ1 (Fℓ (M, ∆2 )) < 1 and ∆1 ∈ B∆1 , so I − Fℓ (M, ∆2 ) ∆1 must be nonsingular. Therefore, I − M ∆ is nonsingular and, by definition, µ∆ (M ) < 1. ⇒ Basically, the argument above is reversed. Again let ∆1 ∈ B∆1 and ∆2 ∈ B∆2 be given, and define ∆ = diag [∆1 , ∆2 ]. Then ∆ ∈ B∆ and, by hypothesis, det (I − M ∆) 6= 0. It is easy to verify from the definition of µ that (always) µ (M ) ≥ max {µ1 (M11 ) , µ2 (M22 )} . We can see that µ2 (M22 ) < 1, which gives that I−M22 ∆2 is also nonsingular. Therefore, the expression in equation (10.16) is valid, giving det (I − M22 ∆2 ) det (I − Fℓ (M, ∆2 ) ∆1 ) = det (I − M ∆) 6= 0. Obviously, I − Fℓ (M, ∆2 ) ∆1 is nonsingular for all ∆i ∈ B∆i , which indicates that the claim is true. ✷ Remark 10.3 This theorem forms the basis for all uses of µ in linear system robustness analysis, whether from a state-space, frequency domain, or Lyapunov approach. ✸ 198 µ AND µ SYNTHESIS The role of the block structure ∆2 in the main loop theorem is clear — it is the structure that the perturbations come from; however, the role of the perturbation structure ∆1 is often misunderstood. Note that µ1 (·) appears on the right-hand side of the theorem, so that the set ∆1 defines what particular property of Fℓ (M, ∆2 ) is considered. As an example, consider the theorem applied with the two simple block structures considered right after Lemma 10.1. Define ∆1 := {δ1 In : δ1 ∈ C}. Hence, for A ∈ Cn×n , µ1 (A) = ρ (A). Likewise, define ∆2 = Cm×m ; then for D ∈ Cm×m , µ2 (D) = σ(D). Now, let ∆ be the diagonal augmentation of these two sets, namely (" ) # δ1 In 0n×m m×m ∆ := ⊂ C(n+m)×(n+m) . : δ1 ∈ C, ∆2 ∈ C 0m×n ∆2 Let A ∈ Cn×n , B ∈ Cn×m , C ∈ Cm×n , and D ∈ Cm×m be given, and interpret them as the state-space model of a discrete time system xk+1 = Axk + Buk yk = Cxk + Duk . Let M ∈ C(n+m)×(n+m) be the block state-space matrix of the system " # A B M= . C D Applying the theorem with these data gives that the following are equivalent: • The spectral radius of A satisfies ρ (A) < 1, and   max σ D + Cδ1 (I − Aδ1 )−1 B < 1. (10.17) • The maximum singular value of D satisfies σ(D) < 1, and   −1 max ρ A + B∆2 (I − D∆2 ) C < 1. (10.18) δ1 ∈C |δ1 |≤1 ∆2 ∈Cm×m σ(∆2 )≤1 • The structured singular value of M satisfies µ∆ (M ) < 1. (10.19) The first condition is recognized by two things: The system is stable, and the || · ||∞ norm on the transfer function from u to y is less than 1 (by replacing δ1 with 1z ):     kGk∞ := max σ D + C (zI − A)−1 B = max σ D + Cδ1 (I − Aδ1 )−1 B . z∈C |z|≥1 δ1 ∈C |δ1 |≤1 10.2. Structured Singular Value 199 −1 The second condition implies that (I − D∆2 ) is well defined for all σ(∆2 ) ≤ 1 and that a robust stability result holds for the uncertain difference equation   xk+1 = A + B∆2 (I − D∆2 )−1 C xk where ∆2 is any element in Cm×m with σ(∆2 ) ≤ 1, but otherwise unknown. This equivalence between the small gain condition, kGk∞ < 1, and the stability robustness of the uncertain difference equation is well-known. This is the small gain theorem, in its necessary and sufficient form for linear, time invariant systems with one of the components norm bounded, but otherwise unknown. What is important to note is that both of these conditions are equivalent to a condition involving the structured singular value of the state-space matrix. Already we have seen that special cases of µ are the spectral radius and the maximum singular value. Here we see that other important linear system properties — namely, robust stability and input-output gain — are also related to a particular case of the structured singular value. Example 10.3 Let M , ∆1 , and ∆2 be defined as in the beginning of this section. Now suppose µ2 (M22 ) < 1. Find max µ1 (Fℓ (M, ∆2 )) . ∆2 ∈B∆2 This can be done iteratively as follows: max µ1 (Fℓ (M, ∆2 )) = α ∆2 ∈B∆2 !! # M11 /α M12 /α =1 , ∆2 ⇐⇒ max µ1 Fℓ ∆2 ∈B∆2 M21 M22 " #! M11 /α M12 /α = 1. ⇐⇒ µ∆ M21 M22 " Hence max µ1 (Fℓ (M, ∆2 )) = ∆2 ∈B∆2 ( α: µ∆ " M11 /α M12 /α M21 M22 For example, let ∆1 = δI2 , ∆2 ∈ C2×2 : " # " # " 0.1 0.2 1 0 1 A= , B= , C= 1 0 1 1 1 2 3 # , D= Find αmax = sup σ(∆2 )≤1 ρ(A + B∆2 (I − D∆2 )−1 C). " #! ) =1 . 0.5 0 0 0.8 # . 200 µ AND µ SYNTHESIS Define ∆ = " δI2 ∆2 # αmax = . Then a bisection search can be done to find ( α: µ∆ " A/α B/α C D #! =1 ) = 21.77. Related MATLAB Commands: unwrapp, muunwrap, dypert, sisorat 10.3 Structured Robust Stability and Performance 10.3.1 Robust Stability The most well-known use of µ as a robustness analysis tool is in the frequency domain. Suppose G(s) is a stable, real rational, multi-input, multioutput transfer function of a linear system. For clarity, assume G has q1 inputs and p1 outputs. Let ∆ be a block structure, as in equation (10.1), and assume that the dimensions are such that ∆ ⊂ Cq1 ×p1 . We want to consider feedback perturbations to G that are themselves dynamical systems with the block diagonal structure of the set ∆. Let M (∆) denote the set of all block diagonal and stable rational transfer functions that have block structures such as ∆.  M (∆) := ∆(·) ∈ RH∞ : ∆(so ) ∈ ∆ for all so ∈ C+ Theorem 10.7 Let β > 0. The loop shown below is well-posed and internally stable for all ∆(·) ∈ M (∆) with k∆k∞ < β1 if and only if sup ω∈R w1 e1 ✲e + ✻ + µ∆ (G(jω)) ≤ β. ✲ ∆ G(s) ✛ Proof. (⇐=) Suppose sups∈C+ e2 + w2 ❄ e✛+ µ∆ (G(s)) ≤ β. Then det(I − G(s)∆(s)) 6= 0 for all s ∈ C+ ∪ {∞} whenever k∆k∞ < 1/β (i.e., the system is robustly stable). Now it is sufficient to show that sup s∈C+ µ∆ (G(s)) = sup µ∆ (G(jω)). ω∈R 10.3. Structured Robust Stability and Performance 201 It is clear that sup µ∆ (G(s)) = sup µ∆ (G(s)) ≥ sup µ∆ (G(jω)). s∈C+ ω s∈C+ Now suppose sups∈C+ µ∆ (G(s)) > β; then by the definition of µ, there is an so ∈ C+ ∪ {∞} and a complex structured ∆ such that σ(∆) < 1/β and det(I − G(so )∆) = 0. This implies that there is a 0 ≤ ω̂ ≤ ∞ and 0 < α ≤ 1 such that det(I − G(j ω̂)α∆) = 0. This, in turn, implies that µ∆ (G(j ω̂)) > β since σ(α∆) < 1/β. In other words, sups∈C+ µ∆ (G(s)) ≤ supω µ∆ (G(jω)). The proof is complete. (=⇒) Suppose supω∈R µ∆ (G(jω)) > β. Then there is a 0 < ωo < ∞ such that µ∆ (G(jωo )) > β. By Remark 10.1, there is a complex ∆c ∈ ∆ that each full block has rank 1 and σ(∆c ) < 1/β such that I − G(jωo )∆c is singular. Next, using the same construction used in the proof of the small gain theorem (Theorem 8.1), one can find a rational ∆(s) such that k∆(s)k∞ = σ(∆c ) < 1/β, ∆(jωo ) = ∆c , and ∆(s) destabilizes the system. ✷ Hence, the peak value on the µ plot of the frequency response determines the size of perturbations that the loop is robustly stable against. Remark 10.4 The internal stability with a closed ball of uncertainties is more complicated. The following example is shown in Tits and Fan [1995]. Consider " # 0 −1 1 G(s) = s+1 1 0 and ∆ = δ(s)I2 . Then sup ω∈R µ∆ (G(jω)) = sup ω∈R 1 = µ∆ (G(j0)) = 1. |jω + 1| On the other hand, µ∆ (G(s)) < 1 for all s 6= 0, s ∈ C+ , and the only matrices in the form of Γ = γI2 with |γ| ≤ 1 for which det(I − G(0)Γ) = 0 are the complex matrices ±jI2 . Thus, clearly, (I − G(s)∆(s))−1 ∈ RH∞ for all real rational ∆(s) = δ(s)I2 with kδk∞ ≤ 1 since ∆(0) must be real. This shows that supω∈R µ∆ (G(jω)) < 1 is not necessary for (I − G(s)∆(s))−1 ∈ RH∞ with the closed ball of structured uncertainty k∆k∞ ≤ 1. Similar examples with no repeated blocks are 1 M , where M is any real matrix with µ∆ (M ) = 1 for generated by setting G(s) = s+1 which there is no real ∆ ∈ ∆ with σ(∆) = 1 such that det(I − M ∆) = 0. For example, let      " # δ1   0 β      −β α α  , ∆=  M = γ α  δ2  , δi ∈ C   0 −γ γ   δ3 γ −α 202 µ AND µ SYNTHESIS with γ 2 = 21 and β 2 + 2α2 = 1. Then it is shown in Packard and Doyle [1993] that µ∆ (M ) = 1 and all ∆ ∈ ∆ with σ(∆) = 1 that satisfy det(I − M ∆) = 0 must be complex. ✸ Remark 10.5 Let ∆ ∈ RH∞ be a structured uncertainty and # " G11 (s) G12 (s) G(s) = ∈ RH∞ . G21 (s) G22 (s) Then Fu (G, ∆) ∈ RH∞ does not necessarily imply (I − G11 ∆)−1 ∈ RH∞ whether ∆ is in an open ball or is in a closed ball. For example, consider  1  0 1 s+1   10 G(s) =  0 0  s+1 1 0 0 # " δ1 1 with k∆k∞ < 1. Then Fu (G, ∆) = and ∆ = 1 ∈ RH∞ for all 1 − δ1 s+1 δ2 admissible ∆ (k∆k∞ < 1) but (I − G11 ∆)−1 ∈ RH∞ is true only for k∆k∞ < 0.1. ✸ 10.3.2 Robust Performance Often, stability is not the only property of a closed-loop system that must be robust to perturbations. Typically, there are exogenous disturbances acting on the system (wind gusts, sensor noise) that result in tracking and regulation errors. Under perturbation, the effect that these disturbances have on error signals can greatly increase. In most cases, long before the onset of instability, the closed-loop performance will degrade to the point of unacceptability (hence the need for a “robust performance” test). Such a test will indicate the worst-case level of performance degradation associated with a given level of perturbations. Assume Gp is a stable, real-rational, proper transfer function with q1 + q2 inputs and p1 + p2 outputs. Partition Gp in the obvious manner # " G11 G12 Gp (s) = G21 G22 so that G11 has q1 inputs and p1 outputs, and so on. Let ∆ ⊂ Cq1 ×p1 be a block structure, as in equation (10.1). Define an augmented block structure: ) (" # ∆ 0 q2 ×p2 . ∆P := : ∆ ∈ ∆, ∆f ∈ C 0 ∆f The setup is to address theoretically the robust performance questions about the following loop: 10.3. Structured Robust Stability and Performance 203 ✲ ∆(s) z ✛ Gp (s) ✛ ✛ w The transfer function from w to z is denoted by Fu (Gp , ∆). Theorem 10.8 Let β > 0. For all ∆(s) ∈ M (∆) with k∆k∞ < β1 , the loop shown above is well-posed, internally stable, and kFu (Gp , ∆)k∞ ≤ β if and only if sup µ∆P (Gp (jω)) ≤ β. ω∈R Note that by internal stability, supω∈R µ∆ (G11 (jω)) ≤ β, then the proof of this theorem is exactly along the lines of the earlier proof for Theorem 10.7, but also appeals to Theorem 10.6. This is a remarkably useful theorem. It says that a robust performance problem is equivalent to a robust stability problem with augmented uncertainty ∆, as shown in Figure 10.5. ✲ ∆f ✲ ∆ ✛ Gp (s) ✛ Figure 10.5: Robust performance vs robust stability Example 10.4 We shall consider again the HIMAT problem from Example 9.1. Use the Simulink block diagram in Example 9.1 and run the following commands to get an interconnection model Ĝ, an H∞ stabilizing controller K and a closed-loop transfer matrix Gp (s) = Fℓ (Ĝ, K). (Do not bother to figure out how hinfsyn works; it will be considered in detail in Chapter 14.) 204 µ AND µ SYNTHESIS ≫ [A, B, C, D] = linmod(′ aircraft′ ) ≫ Ĝ = pck(A, B, C, D); ≫ [K, Gp , γ] = hinfsyn(Ĝ, 2, 2, 0, 10, 0.001, 2); which gives γ = 1.8612 = kGp k∞ , a stabilizing controller K, and a closed loop transfer matrix Gp :   p 1      p2  z1   # "    d   z2  G G   p11 p12 1   .  , Gp (s) =  e  = Gp (s)   d2  Gp21 Gp22  1     n  e2  1  n2 2 maximum singular value 1.5 1 0.5 0 −3 10 −2 −1 10 10 0 10 frequency (rad/sec) 1 10 2 10 Figure 10.6: Singular values of Gp (jω) Now generate the singular value frequency responses of Gp : ≫ w=logspace(-3,3,300); ≫ Gpf = frsp(Gp , w); ≫ [u, s, v] = vsvd(Gpf ); % Gpf is the frequency response of Gp ; 3 10 10.3. Structured Robust Stability and Performance 205 ≫ vplot(′ liv, m′ , s) The singular value frequency responses of Gp are shown in Figure 10.6. To test the robust stability, we need to compute kGp11 k∞ : ≫ Gp11 = sel(Gp , 1 : 2, 1 : 2); ≫ norm of Gp11 = hinfnorm(Gp11 , 0.001); which gives kGP 11 k∞ = 0.933 < 1. So the system is robustly stable. To check the robust performance, we shall compute the µ∆P (Gp (jω)) for each frequency with " # ∆ ∆P = , ∆ ∈ C2×2 , ∆f ∈ C4×2 . ∆f Maximum Singular Value and mu 2 maximum singular value 1.5 1 mu bounds 0.5 −3 10 −2 10 −1 10 0 10 frequency (rad/sec) 1 10 Figure 10.7: µ∆P (Gp (jω)) and σ(Gp (jω)) ≫ blk=[2,2;4,2]; ≫ [bnds,dvec,sens,pvec]=mu(Gpf,blk); ≫ vplot(′ liv, m′ , vnorm(Gpf ), bnds) ≫ title(′ Maximum Singular Value and mu′ ) ≫ xlabel(′ frequency(rad/sec)′ ) 2 10 3 10 206 µ AND µ SYNTHESIS ≫ text(0.01, 1.7,′ maximum singular value′ ) ≫ text(0.5, 0.8,′ mu bounds′ ) The structured singular value µ∆P (Gp (jω)) and σ(Gp (jω)) are shown in Figure 10.7. It is clear that the robust performance is not satisfied. Note that " #! Gp11 Gp12 ≤ 1. max kFu (Gp , ∆)k∞ ≤ γ ⇐⇒ sup µ∆P k∆k∞ ≤1 ω Gp21 /γ Gp22 /γ Using a bisection algorithm, we can also find the worst performance: max kFu (Gp , ∆)k∞ = 12.7824. k∆k∞ ≤1 10.3.3 Two-Block µ: Robust Performance Revisited Suppose that the uncertainty block is given by # " ∆1 ∈ RH∞ ∆= ∆2 with k∆k∞ < 1 and that the interconnection model G is given by " # G11 (s) G12 (s) G(s) = ∈ RH∞ . G21 (s) G22 (s) Then the closed-loop system is well-posed and internally stable iff supω µ∆ (G(jω)) ≤ 1. Let " # dω I Dω = , dω ∈ R+ . I Then Dω G(jω)Dω−1 = " G11 (jω) dω G12 (jω) 1 G (jω) G22 (jω) dω 21 # . Hence, by Theorem 10.4, at each frequency ω #! " G11 (jω) dω G12 (jω) . µ∆ (G(jω)) = inf σ 1 dω ∈R+ G22 (jω) dω G21 (jω) (10.20) Since the minimization is convex in log dω (see, Doyle [1982]), the optimal dω can be found by a search; however, two approximations to dω can be obtained easily by approximating the right-hand side of equation (10.20): 10.3. Structured Robust Stability and Performance 207 (1) Note that µ∆ (G(jω)) ≤ inf σ dω ∈R+ s " kG11 (jω)k dω kG12 (jω)k 1 kG22 (jω)k dω kG21 (jω)k #!   1 2 2 2 2 kG11 (jω)k + d2ω kG12 (jω)k + 2 kG21 (jω)k + kG22 (jω)k dω ∈R+ dω q 2 2 = kG11 (jω)k + kG22 (jω)k + 2 kG12 (jω)k kG21 (jω)k ≤ inf with the minimizing dω given by  q kG21 (jω)k    kG12 (jω)k dˆω = 0    ∞ if G12 6= 0 & G21 6= 0, (10.21) if G21 = 0, if G12 = 0. (2) Alternative approximation can be obtained by using the Frobenius norm: " # G11 (jω) dω G12 (jω) µ∆ (G(jω)) ≤ inf 1 dω ∈R+ G22 (jω) dω G21 (jω) F = s inf dω ∈R+  kG11 (jω)k2F + d2ω kG12 (jω)k2F + 1 kG21 (jω)k2F + kG22 (jω)k2F d2ω q 2 2 = kG11 (jω)kF + kG22 (jω)kF + 2 kG12 (jω)kF kG21 (jω)kF with the minimizing dω     ˜ dω =    given by q 0 ∞ kG21 (jω)kF kG12 (jω)kF if G12 6= 0 & G21 6= 0, if G21 = 0, if G12 = 0.  (10.22) It can be shown that the approximations for the scalar dω obtained previously are exact for a 2 × 2 matrix G. For higher dimensional G, the approximations for dω are still reasonably good. Hence an approximation of µ can be obtained as #! " G11 (jω) dˆω G12 (jω) (10.23) µ∆ (G(jω)) ≤ σ 1 G21 (jω) G22 (jω) d̂ ω 208 µ AND µ SYNTHESIS or, alternatively, as µ∆ (G(jω)) ≤ σ " G11 (jω) d˜ω G12 (jω) 1 G21 (jω) G22 (jω) d˜ ω #! . (10.24) We can now see how these approximated µ tests are compared with the sufficient conditions obtained in Chapter 8. Example 10.5 Consider again the robust performance problem of a system with output multiplicative uncertainty in Chapter 8 (see Figure 8.10): P∆ = (I + W1 ∆W2 )P, k∆k∞ < 1. Then it is easy to show that the problem can be put in the general framework by selecting # " −W2 To W1 −W2 To Wd G(s) = We So W1 We So Wd and that the robust performance condition is satisfied if and only if kW2 To W1 k∞ ≤ 1 (10.25) kFu (G, ∆)k∞ ≤ 1 (10.26) and for all ∆ ∈ RH∞ with k∆k∞ < 1. But equations (10.25) and (10.26) are satisfied iff for each frequency ω #! " −W2 To W1 −dω W2 To Wd ≤ 1. µ∆ (G(jω)) = inf σ 1 dω ∈R+ We So Wd dω We So W1 Note that, in contrast to the sufficient condition obtained in Chapter 8, this condition is an exact test for robust performance. To compare the µ test with the criteria obtained in Chapter 8, some upper-bounds for µ can be derived. Let s kWe So W1 k dω = . kW2 To Wd k Then, using the first approximation for µ, we get q 2 2 kW2 To W1 k + kWe So Wd k + 2 kW2 To Wd k kWe So W1 k µ∆ (G(jω)) ≤ q 2 2 ≤ kW2 To W1 k + kWe So Wd k + 2κ(W1−1 Wd ) kW2 To W1 k kWe So Wd k ≤ kW2 To W1 k + κ(W1−1 Wd ) kWe So Wd k 10.3. Structured Robust Stability and Performance 209 where W1 is assumed to be invertible in the last two inequalities. The last term is exactly the sufficient robust performance criteria obtained in Chapter 8. It is clear that any term preceding the last forms a tighter test since κ(W1−1 Wd ) ≥ 1. Yet another alternative sufficient test can be obtained from the preceding sequence of inequalities: q µ∆ (G(jω)) ≤ κ(W1−1 Wd )(kW2 To W1 k + kWe So Wd k). Note that this sufficient condition is not easy to get from the approach taken in Chapter 8 and is potentially less conservative than the bounds derived there. Next we consider the skewed specification problem, but first the following lemma is needed in the sequel. Lemma 10.9 Suppose σ = σ1 ≥ σ2 ≥ . . . ≥ σm = σ > 0, then   σ 1 σ 2 inf max (dσi ) + + = . d∈R+ i (dσi )2 σ σ Proof. Consider a function y = x + 1/x; then y is a convex function and the maximization over a closed interval is achieved at the boundary of the interval. Hence for any fixed d     1 1 1 2 2 + + = max (dσ) . , (dσ) max (dσi )2 + i (dσi )2 (dσ)2 (dσ)2 Then the minimization over d is obtained iff (dσ)2 + which gives d2 = 1 σ σ. 1 1 = (dσ)2 + , 2 (dσ) (dσ)2 The result then follows from substituting d. ✷ Example 10.6 As another example, consider again the skewed specification problem from Chapter 8. Then the corresponding G matrix is given by # " −W2 Ti W1 −W2 KSo Wd . G= We So P W1 We So Wd So the robust performance specification is satisfied iff #! " −W2 Ti W1 −dω W2 KSo Wd µ∆ (G(jω)) = inf σ ≤1 1 dω ∈R+ We So Wd dω We So P W1 210 µ AND µ SYNTHESIS for all ω ≥ 0. As in the last example, an upper-bound can be obtained by taking s kWe So P W1 k . dω = kW2 KSo Wd k Then q µ∆ (G(jω)) ≤ κ(Wd−1 P W1 )(kW2 Ti W1 k + kWe So Wd k). In particular, this suggests that the robust performance margin is inversely proportional to the square root of the plant condition number if Wd = I and W1 = I. This can be further illustrated by considering a plant-inverting control system. To simplify the exposition, we shall make the following assumptions: We = ws I, Wd = I, W1 = I, W2 = wt I, and P is stable and has a stable inverse (i.e., minimum phase) (P can be strictly proper). Furthermore, we shall assume that the controller has the form K(s) = P −1 (s)l(s) where l(s) is a scalar loop transfer function that makes K(s) proper and stabilizes the closed loop. This compensator produces diagonal sensitivity and complementary sensitivity functions with identical diagonal elements; namely, So = Si = 1 I, 1 + l(s) To = T i = l(s) I. 1 + l(s) Denote l(s) 1 , τ (s) = 1 + l(s) 1 + l(s) and substitute these expressions into G; we get # " −wt τ I −wt τ P −1 . G= ws εP ws εI ε(s) = The structured singular value for G at frequency ω can be computed by " #! −wt τ I −wt τ (dP )−1 µ∆ (G(jω)) = inf σ . d∈R+ ws εdP ws εI Let the singular value decomposition of P (jω) at frequency ω be P (jω) = U ΣV ∗ , Σ = diag(σ1 , σ2 , . . . , σm ) with σ1 = σ and σm = σ, where m is the dimension of P . Then " #! −wt τ I −wt τ (dΣ)−1 µ∆ (G(jω)) = inf σ d∈R+ ws εdΣ ws εI 10.3. Structured Robust Stability and Performance 211 since unitary operations do not change the singular values of a matrix. Note that " # −wt τ I −wt τ (dΣ)−1 = P1 diag(M1 , M2 , . . . , Mm )P2 ws εdΣ ws εI where P1 and P2 are permutation matrices and where " # −wt τ −wt τ (dσi )−1 Mi = . ws εdσi ws ε Hence #! −wt τ (dσi )−1 inf max σ d∈R+ i ws ε " ! # i h −wt τ −1 inf max σ 1 (dσi ) d∈R+ i ws εdσi p inf max (1 + |dσi |−2 )(|ws εdσi |2 + |wt τ |2 ) d∈R+ i s wt τ inf max |ws ε|2 + |wt τ |2 + |ws εdσi |2 + d∈R+ i dσi " µ∆ (G(jω)) = = = = −wt τ ws εdσi 2 . Using Lemma 10.9, it is easy to show that the maximum is achieved at either σ or σ and that optimal d is given by |wt τ | , d2 = |ws ε|σσ so the structured singular value is µ∆ (G(jω)) = s |ws ε|2 + |wt τ |2 + |ws ε||wt τ |[κ(P ) + 1 ]. κ(P ) (10.27) Note that if |ws ε| and |wt τ | are not too large, which is guaranteed if the nominal performance and robust stability conditions are satisfied, then the structured singular value is proportional to the square root of the plant condition number: p µ∆ (G(jω)) ≈ |ws ε||wt τ |κ(P ) . (10.28) This confirms our intuition that an ill-conditioned plant with skewed specifications is hard to control. 212 10.3.4 µ AND µ SYNTHESIS Approximation of Multiple Full Block µ The approximations given in the last subsection can be generalized to the multiple-block µ problem by assuming that M is partitioned consistently with the structure of ∆ = diag(∆1 , ∆2 , . . . , ∆F ) so that  M11 M12   M21 M =  ..  . M22 .. . MF 2 MF 1 and ··· M1F ··· M2F .. . · · · MF F       D = diag(d1 I, . . . , dF −1 I, I). Now DM D  di , dF := 1. = Mij dj  −1 Hence   di µ∆ (M ) ≤ inf σ(DM D ) = inf σ Mij D∈D D∈D dj v u F F   2 uX X di 2 d kMij k i2 ≤ inf σ kMij k ≤ inf t D∈D D∈D dj dj i=1 j=1 v u F F 2 uX X 2 d kMij kF 2i . ≤ inf t D∈D dj i=1 j=1 −1 An approximate D can be found by solving the following minimization problem: inf D∈D F F X X i=1 j=1 2 kMij k d2i d2j or, more conveniently, by minimizing inf D∈D F X F X i=1 j=1 2 kMij kF d2i d2j with dF = 1. The optimal di minimizing the preceding two problems satisfies, respectively, P 2 2 i6=k kMik k di 4 dk = P , k = 1, 2, . . . , F − 1 (10.29) 2 2 j6=k kMkj k /dj 10.4. Overview of µ Synthesis and d4k P =P 213 2 i6=k kMik kF d2i 2 j6=k kMkj kF /d2j , k = 1, 2, . . . , F − 1. (10.30) Using these relations, dk can be obtained by iterations. Example 10.7 Consider a 3 × 3 complex matrix  1+j  M =  5j −2  −20j  −1 + 3j  4−j 10 − 2j 3+j j with structured ∆ = diag(δ1 , δ2 , δ3 ). The largest singular value of M is σ(M ) = 22.9094 and the structured singular value of M computed using the µ Analysis and Synthesis Toolbox is equal to its upper-bound: µ∆ (M ) = inf σ(DM D−1 ) = 11.9636 D∈D with the optimal scaling Dopt = diag(0.3955, 0.6847, 1). The optimal D minimizing inf D∈D F X F X i=1 j=1 2 kMij k d2i d2j is Dsubopt = diag(0.3212, 0.4643, 1), which is solved from equation (10.29). Using this Dsubopt , we obtain another upper-bound for the structured singular value: −1 µ∆ (M ) ≤ σ(Dsubopt M Dsubopt ) = 12.2538. One may also use this Dsubopt as an initial guess for the exact optimization. 10.4 Overview of µ Synthesis This section briefly outlines various synthesis methods. The details are somewhat complicated and are treated in the other parts of this book. At this point, we simply want to point out how the analysis theory discussed in the previous sections leads naturally to synthesis questions. From the analysis results, we see that each case eventually leads to the evaluation of kM kα α = 2, ∞, or µ (10.31) 214 µ AND µ SYNTHESIS for some transfer matrix M . Thus when the controller is put back into the problem, it involves only a simple linear fractional transformation, as shown in Figure 10.8, with M = Fℓ (G, K) = G11 + G12 K(I − G22 K)−1 G21 " # G11 G12 where G = is chosen, respectively, as G21 G22 # " P22 P23 • nominal performance only (∆ = 0): G = P32 P33 # " P11 P13 • robust stability only: G = P31 P33  P11  • robust performance: G = P =  P21 P31 z ✛ G P12 P22 P32 ✛ ✛  P13  P23 . P33 w ✲ K Figure 10.8: Synthesis framework Each case then leads to the synthesis problem min kFℓ (G, K)kα for α = 2, ∞, or µ, K (10.32) which is subject to the internal stability of the nominal. The solutions of these problems for α = 2 and ∞ are the focus of the rest of this book. The α = 2 case was already known in the 1960s, and the result presented in this book is simply a new interpretation. The two Riccati solutions for the α = ∞ case were new products of the late 1980s. The synthesis for the α = µ case is not yet fully solved. Recall that µ may be obtained by scaling and applying k·k∞ (for F ≤ 3 and S = 0); a reasonable approach is to “solve” min inf DFℓ (G, K)D−1 ∞ (10.33) −1 K D,D ∈H∞ by iteratively solving for K and D. This is the so-called D-K iteration. The stable and minimum phase scaling matrix D(s) is chosen such that D(s)∆(s) = ∆(s)D(s). [Note 10.4. Overview of µ Synthesis 215 that D(s) is not necessarily belonging to D since D(s) is not necessarily Hermitian, see Remark 10.2.] For a fixed scaling transfer matrix D, minK DFℓ (G, K)D−1 ∞ is a standard H∞ optimization problem that will be solved later in this book. For a given stabilizing controller K, inf D,D−1 ∈H∞ DFℓ (G, K)D−1 ∞ is a standard convex optimization problem and it can be solved pointwise in the frequency domain:   sup inf σ Dω Fℓ (G, K)(jω)Dω−1 . ω Dω ∈D Indeed, inf D,D−1 ∈H∞ DFℓ (G, K)D−1 ∞   = sup inf σ Dω Fℓ (G, K)(jω)Dω−1 . ω Dω ∈D This follows intuitively from the following arguments: The left-hand side is always no smaller than the right-hand side, and, on the other hand, given Dω ∈ D, there is always a real-rational function D(s), stable with stable inverse, such that the Hermitian positive definite factor in the polar decomposition of D(jω) uniformly approximates Dω over ω in R. In particular, in the case of scalar blocks, the magnitude |D(jω)| uniformly approximates Dω over R. Note that when S = 0 (no scalar blocks), ω Dω = diag(dω 1 I, . . . , dF −1 I, I) ∈ D, which is a block diagonal scaling matrix applied pointwise across frequency to the frequency response Fℓ (G, K)(jω). ✛ D ✛ ✛ G ✛ D−1 ✛ ✲ K Figure 10.9: µ synthesis via scaling D-K iterations proceed by performing this two-parameter minimization in sequential fashion: first minimizing over K with D fixed, then minimizing pointwise over D with K fixed, then again over K, and again over D, etc. Details of this process are summarized in the following steps: (i) Fix an initial estimate of the scaling matrix Dω ∈ D pointwise across frequency. (ii) Find scalar transfer functions di (s), d−1 i (s) ∈ RH∞ for i = 1, 2, . . . , (F − 1) such that |di (jω)| ≈ dω . This step can be done using the interpolation theory (Youla i and Saito [1967]); however, this will usually result in very high-order transfer functions, which explains why this process is currently done mostly by graphical matching using lower-order transfer functions. 216 µ AND µ SYNTHESIS (iii) Let D(s) = diag(d1 (s)I, . . . , dF −1 (s)I, I). Construct a state-space model for system " # " # D(s) D−1 (s) , Ĝ(s) = G(s) I I as shown in Figure 10.9. (iv) Solve an H∞ -optimization problem to minimize Fℓ (Ĝ, K) ∞ over all stabilizing K’s. Note that this optimization problem uses the scaled version of G. Let its minimizing controller be denoted by K̂. (v) Minimize σ[Dω Fℓ (G, K̂)Dω−1 ] over Dω , pointwise across frequency.1 Note that this evaluation uses the minimizing K̂ from the last step, but that G is not scaled. The minimization itself produces a new scaling function. Let this new function be denoted by D̂ω . (vi) Compare D̂ω with the previous estimate Dω . Stop if they are close, but otherwise replace Dω with D̂ω and return to step (ii). With either K or D fixed, the global optimum in the other variable may be found using the µ and H∞ solutions. Although the joint optimization of D and K is not convex and the global convergence is not guaranteed, many designs have shown that this approach works very well (see, e.g., Balas [1990]). In fact, this is probably the most effective design methodology available today for dealing with such complicated problems. Detailed treatment of µ analysis is given in Packard and Doyle [1993]. The rest of this book will focus on the H∞ optimization, which is a fundamental tool for µ synthesis. Related MATLAB Commands: musynfit, musynflp, muftbtch, dkit 10.5 Notes and References This chapter is partially based on the lecture notes given by Doyle [1984] and partially based on the lecture notes by Packard [1991] and the paper by Doyle, Packard, and Zhou [1991]. Parts of Section 10.3.3 are based on the paper by Stein and Doyle [1991]. The small µ theorem for systems with nonrational plants and uncertainties is proven in Tits [1995]. Connections were established in Poolla and Tikku [1995] between the 1 The approximate solutions given in the preceding section may be used. 10.6. Problems 217 frequency-dependent D-scaled upper bounds of the structured singular value and the robust performance of a system with arbitrarily slowly varying structured linear perturbations. Robust performance of systems with structured time-varying perturbations was also considered in Shamma [1994] using the constant D-scaled upper bounds of the structured singular value. Other results on µ can be found in Fan and Tits [1986], Fan, Tits, and Doyle [1991], Packard and Doyle [1993], Packard and Pandey [1993], Young [1993], and references therein. 10.6 Problems Problem 10.1 Let M and N be suitably dimensioned matrices and let ∆ be a structured uncertainty. Prove or disprove (a) µ∆ (M ) = 0 =⇒ M = 0; (b) µ∆ (M1 + M2 ) ≤ µ∆ (M1 ) + µ∆ (M2 ). (c) µ∆ (αM ) = |α|µ∆ (M ). (d) µ∆ (I) = 1. (e) µ∆ (M N ) ≤ σ(M )µ∆ (N ). (f) µ∆ (M N ) ≤ σ(N )µ∆ (M ). " # ∆1 0 Problem 10.2 Let ∆ = , where ∆i are structured uncertainties. Show 0 ∆2 " #! M11 M12 that µ∆ = max{µ∆1 (M11 ), µ∆2 (M22 )}. 0 M22 Problem 10.3 Matlab exercise. Let M be a 7 × 7 random real matrix. Take the perturbation structure to be    0     δ1 I3 0   2×2 . ∆=  0 ∆2 0  : δ1 , δ3 ∈ C, ∆2 ∈ C     0 0 δ3 I2 Compute µ(M ) and a singularizing perturbation. # " " ∆1 0 M12 be a complex matrix and let ∆ = Problem 10.4 Let M = M21 0 Show that p µ∆ (M ) = σ(M12 )σ(M21 ). ∆2 # . 218 µ AND µ SYNTHESIS Problem 10.5 Let M = Show that " M11 M21 M12 M22 # be a complex matrix and let ∆ = " ∆1 ∆2 # p σ(M12 )σ(M21 ) − max{σ(M11 ), σ(M22 )} ≤ µ∆ (M ) p ≤ σ(M12 )σ(M21 ) + max{σ(M11 ), σ(M22 )}. Problem 10.6 Let ∆ be all diagonal full blocks and M be partitioned as M = [Mij ], where Mij are matrices with suitable dimensions. Show that µ∆ (M ) ≤ π ([kMij k]) (= ρ ([kMij k])) where π(·) denotes the Perron eigenvalue. Problem 10.7 Show inf D∈Cn×n DM D−1 p = ρ(M ) where k·kp is the induced p-norm, 1 ≤ p ≤ ∞. Problem 10.8 Let D be a nonsingular diagonal matrix D = diag(d1 , d2 , . . . , dn ). Show inf DM D−1 D p = π(M ) if either M = [|mi j|] and 1 ≤ p ≤ ∞ or p = 1 or ∞. Moreover, the optimal D is given by 1/p 1/q 1/p 1/q D = diag(y1 /x1 , y2 /x2 , . . . , yn1/p /x1/q n ) where 1/p + 1/q = 1 and [|mij |]x = π(M )x, y T [|mij |] = π(M )y T . Problem 10.9 Let ∆ be a structured uncertainty defined in the book. Suppose M = xy ∗ with x, y ∈ Cn . Derive an exact expression for µ∆ (M ) in terms of the components of x and y. [Note that ∆ = diag(δ1 I, δ2 I, . . . , δm I, ∆1 , . . . , ∆F ) and µ∆ (M ) = maxU∈U ρ(M U ) = max∆∈B∆ ρ(M ∆).] ∞ ∞ Problem 10.10 Let {xk }∞ k=0 , {zk }k=0 , and {dk }k=0 be sequences satisfying 2 2 2 2 kxk+1 k + kzk k ≤ β 2 (kxk k + kdk k ) for some β < 1 and all k ≥ 0. If d ∈ ℓ2 , show that both x ∈ ℓ2 and z ∈ ℓ2 and the norms are bounded by 2 2 2 2 kzk2 + (1 − β 2 ) kxk2 ≤ β 2 kdk2 + kx0 k Give a system interpretation of this result. . 10.6. Problems 219 Problem 10.11 Consider a SISO feedback system shown here with P = P0 (1 + W1 ∆1 ) + W2 ∆2 , ∆i ∈ RH∞ , k∆i k∞ < 1, i = 1, 2. Suppose W1 and W2 are stable, and P and P0 have the same number of poles in Re{s} > 0. d ❄ W3 e ✲ K ✻ − z ✲ ✲❄ e ✲ P (a) Show that the feedback system is robustly stable if and only if K stabilizes P0 and k |W1 T | + |W2 KS| k∞ ≤ 1 where S= P0 K 1 , T = . 1 + P0 K 1 + P0 K (b) Show that the feedback system has robust performance; that is, kTzd k∞ ≤ 1, if and only if K stabilizes P0 and k |W3 S| + |W1 T | + |W2 KS| k∞ ≤ 1. Problem 10.12 In Problem 10.11, find a matrix # " M11 M12 M= M21 M22 such that z = Fu (M, ∆)d, ∆s = " ∆1 ∆2 # . Assume that K stabilizes P0 . Show that at each frequency µ∆s (M11 ) = inf σ(Ds M11 Ds−1 ) = |W1 T | + |W2 KS| Ds ∈Ds with Ds = " d 1 # . 220 µ AND µ SYNTHESIS Next let ∆ = " ∆s ∆p #   d1  and D =  d2 1  . Show that µ∆ (M ) = inf σ(DM D−1 ) = |W3 S| + |W1 T | + |W2 KS| D∈D Problem 10.13 Let ∆ = diag(∆1 , . . . , ∆F ) be a structured uncertainty and suppose M = xy ∗ with x, y ∈ Cn . Let x and y be partitioned compatibly with the ∆:     y1 x1  x2   y2     x=  ...  , y =  ..  . . xm+F ym+F Show that µ∆ (M ) = inf σ(DM D−1 ) D∈D and the minimizing di are given by d2i = kyi k kxF k . kxi k kyF k Problem 10.14 Consider M ∈ C2n×2n to be given. Let ∆2 be a n × n block structure, and suppose that µ2 (M22 ) < 1. Suppose also that Fl (M, ∆2 ) is invertible for all ∆2 ∈ ˆ such that B∆2 . For each α > 1, find a matrix Wα and a block structure ∆ max κ(Fl (M, ∆2 )) < α ∆2 ∈B∆2 if and only if µ∆ ˆ (Wα ) < 1 where κ is the condition number. Problem 10.15 Let G(s) ∈ RH∞ be an m × m symmetric transfer matrix, i.e., GT (s) = G(s), and let ∆(s) ∈ RH∞ be a diagonal perturbation, i.e.,   δ (s) 1  ∆(s) =   δ2 (s) .. . δm (s)  .  Show (I − G∆)−1 ∈ RH∞ for all k∆k∞ ≤ γ if and only if kGk∞ < 1/γ. [Hint: Note that for a complex symmetric matrix M = M T ∈ Cm×m , there is a unitary matrix U and a diagonal matrix Σ = diag(σ1 , σ2 , . . . , σm ) ≥ 0 such that M = U ΣU T , see Horn and Johnson [1990, page 204]. For detailed discussion of this problem, see Qiu [1995].] Chapter 11 Controller Parameterization The basic configuration of the feedback systems considered in this chapter is an LFT , as shown in Figure 11.1, where G is the generalized plant with two sets of inputs: the exogenous inputs w, which include disturbances and commands, and control inputs u. The plant G also has two sets of outputs: the measured (or sensor) outputs y and the regulated outputs z. K is the controller to be designed. A control problem in this setup is either to analyze some specific properties (e.g., stability or performance) of the closed loop or to design the feedback control K such that the closed-loop system is stable in some appropriate sense and the error signal z is specified (i.e., some performance specifications are satisfied). In this chapter we are only concerned with the basic internal stabilization problems. We will see again that this setup is very convenient for other general control synthesis problems in the coming chapters. ✛z ✛w G y ✲ K ✛ u Figure 11.1: General system interconnection Suppose that a given feedback system is feedback stabilizable. In this chapter, the problem we are mostly interested in is parameterizing all controllers that stabilize the system. The parameterization of all internally stabilizing controllers was first introduced by Youla et al. [1976a, 1976b] using the coprime factorization technique. We shall, however, focus on the state-space approach in this chapter. 221 222 11.1 CONTROLLER PARAMETERIZATION Existence of Stabilizing Controllers Consider a system described by the standard block diagram in Figure 11.1. Assume that G(s) has a stabilizable and detectable realization of the form G(s) = " G11 (s) G12 (s) G21 (s) G22 (s) #   A B1 B2  =  C1 C2 D11 D21  D12  . D22 (11.1) The stabilization problem is to find feedback mapping K such that the closed-loop system is internally stable; the well-posedness is required for this interconnection. Definition 11.1 A proper system G is said to be stabilizable through output feedback if there exists a proper controller K internally stabilizing G in Figure 11.1. Moreover, a proper controller K(s) is said to be admissible if it internally stabilizes G. The following result is standard and follows from Chapter 3. Lemma 11.1 There exists a proper K achieving internal stability iff (A, B2 ) is stabilizable and (C2 , A) is detectable. Further, let F and L be such that A + B2 F and A + LC2 are stable; then an observer-based stabilizing controller is given by # " A + B2 F + LC2 + LD22 F −L . K(s) = F 0 Proof. (⇐) By the stabilizability and detectability assumptions, there exist F and L such that A + B2 F and A + LC2 are stable. Now let K(s) be the observer-based controller given in the lemma, then the closed-loop A matrix is given by " # A B2 F à = . −LC2 A + B2 F + LC2 It is easy to check that this matrix is similar to the matrix " # A + LC2 0 . −LC2 A + B2 F Thus the spectrum of à equals the union of the spectra of A + LC2 and A + B2 F . In particular, à is stable. (⇒) If (A, B2 ) is not stabilizable or if (C2 , A) is not detectable, then there are some eigenvalues of à that are fixed in the right-half plane, no matter what the compensator is. The details are left as an exercise. ✷ 11.1. Existence of Stabilizing Controllers 223 The stabilizability and detectability conditions of (A, B2 , C2 ) are assumed throughout the remainder of this chapter.1 It follows that the realization for G22 is stabilizable and detectable, and these assumptions are enough to yield the following result. G22 ✛ y u ✲ K Figure 11.2: Equivalent stabilization diagram " A B2 # for G22 is stabilizable and C2 D22 detectable. Then the system in Figure 11.1 is internally stable iff the one in Figure 11.2 is internally stable. Lemma 11.2 Suppose the inherited realization In other words, K(s) internally stabilizes G(s) if and only if it internally stabilizes G22 [provided that (A, B2 , C2 ) is stabilizable and detectable]. Proof. The necessity follows from the definition. To show the sufficiency, it is sufficient to show that the system in Figure 11.1 and that in Figure 11.2 share the same A matrix, which is obvious. ✷ From Lemma 11.2, we see that the stabilizing controller for G depends only on G22 . Hence all stabilizing controllers for G can be obtained by using only G22 . Remark 11.1 There should be no confusion between a given realization for a transfer matrix G22 and the inherited realization from G, where G22 is a submatrix. A given realization for G22 may be stabilizable and detectable while the inherited realization may not be. For instance, # " −1 1 1 G22 = = s+1 1 0 is a minimal realization but the inherited realization  −1 0 #  "  0 1 G11 G12 =  G21 G22  0 1 1 0 of G22 from  0 1  1 0   0 0   0 0 1 It should be clear that the stabilizability and detectability of a realization for G do not guarantee the stabilizability and/or detectability of the corresponding realization for G22 . 224 CONTROLLER PARAMETERIZATION is G22  −1 0 1   1   = , = 0 1 0  s+1 1 0 0  which is neither stabilizable nor detectable. 11.2 ✸ Parameterization of All Stabilizing Controllers Consider again the standard system block diagram in Figure 11.1 with  A  G(s) =  C1 C2 B1 D11 D21 B2 D12 D22   = " G11 (s) G12 (s) G21 (s) G22 (s) # . Suppose (A, B2 ) is stabilizable and (C2 , A) is detectable. In this section we discuss the following problem: Given a plant G, parameterize all controllers K that internally stabilize G. This parameterization for all stabilizing controllers is usually called Youla parameterization. The parameterization of all stabilizing controllers is easy when the plant itself is stable. Theorem 11.3 Suppose G ∈ RH∞ ; then the set of all stabilizing controllers can be described as K = Q(I + G22 Q)−1 (11.2) for any Q ∈ RH∞ and I + D22 Q(∞) nonsingular. Remark 11.2 This result is very natural considering Corollary 5.3, which says that a controller K stabilizes a stable plant G22 iff K(I − G22 K)−1 is stable. Now suppose Q = K(I −G22 K)−1 is a stable transfer matrix, then K can be solved from this equation which gives exactly the controller parameterization in the preceding theorem. ✸ Proof. Note that G22 (s) is stable by the assumptions on G. Then it is straightforward to verify that the controllers given previously stabilize G22 . On the other hand, suppose K0 is a stabilizing controller; then Q0 := K0 (I − G22 K0 )−1 ∈ RH∞ , so K0 can be expressed as K0 = Q0 (I + G22 Q0 )−1 . Note that the invertibility in the last equation is guaranteed by the well-posedness condition of the interconnected system with controller K0 since I + D22 Q0 (∞) = (I − D22 K0 (∞))−1 . ✷ However, if G is not stable, the parameterization is much more complicated. The results can be more conveniently stated using state-space representations. 11.2. Parameterization of All Stabilizing Controllers 225 Theorem 11.4 Let F and L be such that A + LC2 and A + B2 F are stable; then all controllers that internally stabilize G can be parameterized as the transfer matrix from y to u: ✛y ✛u J  ✛ A + B2 F + LC2 + LD22 F  J = ✲ Q F −(C2 + D22 F ) −L B2 + LD22 0 I I −D22    with any Q ∈ RH∞ and I + D22 Q(∞) nonsingular. Furthermore, the set of all closedloop transfer matrices from w to z achievable by an internally stabilizing proper controller is equal to Fℓ (T, Q) = {T11 + T12 QT21 : Q ∈ RH∞ , I + D22 Q(∞) invertible} where T is given by T = " T11 T12 T21 T22  A + B2 F #   0 =  C +D F 12  1 0 −B2 F A + LC2 B1 B1 + LD21 −D12 F D11 D21 C2  B2  0  . D12   0 Proof. Let K = Fℓ (J, Q). Then it is straightforward to verify, by using the statespace star product formula and some tedious algebra, that Fℓ (G, K) = T11 + T12 QT21 with the T given in the theorem. Hence the controller K = Fℓ (J, Q) for any given Q ∈ RH∞ does internally stabilize G. Now let K be any stabilizing controller for G; ˆ K) ∈ RH∞ , where then Fℓ (J,   A −L B2   Jˆ =  −F 0 I . C2 I D22 (Jˆ is stabilized by K since it has the same G22 matrix as G.) ˆ K) ∈ RH∞ ; then Fℓ (J, Q0 ) = Fℓ (J, Fℓ (J, ˆ K)) =: Fℓ (Jtmp , K), Let Q0 := Fℓ (J, where Jtmp can be obtained by using the state-space star product formula given in Chapter 9:   A + LC2 + B2 F + LD22 F −(B2 + LD22 )F −L B2 + LD22    −L B2 + LD22  L(C2 + D22 F ) A − LD22 F   Jtmp =   F −F 0 I   I 0 −(C2 + D22 F ) C2 + D22 F 226 CONTROLLER PARAMETERIZATION  A + LC2   0 =   0  0 # " 0 I . = I 0 −(B2 + LD22 )F A + B2 F −F C2  −L B2 + LD22   0 0   0 I  I 0 Hence Fℓ (J, Q0 ) = Fℓ (Jtmp , K) = K. This shows that any stabilizing controller can be expressed in the form of Fℓ (J, Q0 ) for some Q0 ∈ RH∞ . ✷ An important point to note is that the closed-loop transfer matrix is simply an affine function of the controller parameter matrix Q. The proper K’s achieving internal stability are precisely those represented in Figure 11.3. ✛ z G w ✛ ✛ y u D22 ✛ ❄ c✛ ❄ c✛ − C2 ✛ R ✛ c✛ c✛ B2 ✛ ✻✻ c✛ ✻ ✲ A ✲ F ✲ −L y1 u1 ✲ Q Figure 11.3: Structure of stabilizing controllers It is interesting to note that the system in the dashed box is an observer-based stabilizing controller for G (or G22 ). Furthermore, it is easy to show that the transfer function between (y, u1 ) and (u, y1 ) is J; that is, " # " # u y =J . y1 u1 It is also easy to show that the transfer matrix from u1 to y1 is T22 = 0. 11.2. Parameterization of All Stabilizing Controllers 227 This diagram of the parameterization of all stabilizing controllers also suggests an interesting interpretation: Every internal stabilization amounts to adding stable dynamics to the plant and then stabilizing the extended plant by means of an observer. The precise statement is as follows: For simplicity of the formulas, only the cases of strictly proper G22 and K are treated. Theorem 11.5 Assume that G22 and K are strictly proper and the system in Figure 11.1 is internally stable. Then G22 can be embedded in a system # " Ae Be Ce where Ae = " A 0 0 Aa # , Be = 0 " B2 0 # , Ce = and where Aa is stable, such that K has the form " Ae + Be Fe + Le Ce K= Fe −Le 0 h C2 # 0 i (11.3) (11.4) where Ae + Be Fe and Ae + Le Ce are stable. Proof. K is representable as in Figure 11.3 for some Q in RH∞ . For K to be strictly proper, Q must be strictly proper. Take a minimal realization of Q: # " Aa Ba . Q= Ca 0 Since Q ∈ RH∞ , Aa is stable. Let x and xa denote state vectors for J and Q, respectively, and write the equations for the system in Figure 11.3: ẋ = (A + B2 F + LC2 )x − Ly + B2 u1 u = F x + u1 y1 ẋa = −C2 x + y = Aa xa + Ba y1 u1 = Ca xa These equations yield ẋe = (Ae + Be Fe + Le Ce )xe − Le y u = Fe xe 228 CONTROLLER PARAMETERIZATION where xe := " x xa # , Fe := h F Ca i , Le := " L −Ba # and where Ae , Be , Ce are as in equation (11.3). ✷ 1 . s−1 We shall find all stabilizing controllers for P such that the steady-state errors with respect to the step input and sin 2t are both zero. It is easy to see that the controller must provide poles at 0 and ±2j. Now let the set of stabilizing controllers for a mod(s + 1)3 ified plant be Km . Then the desired set of controllers is given by (s − 1)s(s2 + 22 ) (s + 1)3 K= Km . s(s2 + 22 ) Example 11.1 Consider a standard feedback system shown in Figure 5.1 with P = 11.3 Coprime Factorization Approach In this section, all stabilizing controller parameterization will be derived using the conventional coprime factorization approach. Readers should be familiar with the results presented in Section 5.4 of Chapter 5 before preceding further. Theorem 11.6 Let G22 = N M −1 = M̃ −1 Ñ be the rcf and lcf of G22 over RH∞ , respectively. Then the set of all proper controllers achieving internal stability is parameterized either by K = (U0 + M Qr )(V0 + N Qr )−1 , det(I + V0−1 N Qr )(∞) 6= 0 (11.5) for Qr ∈ RH∞ or by K = (Ṽ0 + Ql Ñ )−1 (Ũ0 + Ql M̃ ), det(I + Ql Ñ Ṽ0−1 )(∞) 6= 0 for Ql ∈ RH∞ , where U0 , V0 , Ũ0 , Ṽ0 ∈ RH∞ satisfy the Bezout identities: Ṽ0 M − Ũ0 N = I, M̃ V0 − Ñ U0 = I. Moreover, if U0 , V0 , Ũ0 , and Ṽ0 are chosen such that U0 V0−1 = Ṽ0−1 Ũ0 ; that is, # # " #" " I 0 M U0 Ṽ0 −Ũ0 = N V0 0 I −Ñ M̃ (11.6) 11.3. Coprime Factorization Approach 229 then K = (U0 + M Qy )(V0 + N Qy )−1 = (Ṽ0 + Qy Ñ )−1 (Ũ0 + Qy M̃ ) = Fℓ (Jy , Qy ) where Jy := " U0 V0−1 Ṽ0−1 V0−1 −V0−1 N # (11.7) (11.8) and where Qy ranges over RH∞ such that (I + V0−1 N Qy )(∞) is invertible. Proof. We shall prove the parameterization given in equation (11.5) first. Assume that K has the form indicated, and define U := U0 + M Qr , V := V0 + N Qr . Then M̃ V − Ñ U = M̃ (V0 + N Qr ) − Ñ (U0 + M Qr ) = M̃V0 − ÑU0 + (M̃ N − Ñ M )Qr = I. Thus K achieves internal stability by Lemma 5.7. Conversely, suppose K is proper and achieves internal stability. Introduce an rcf of K over RH∞ as K = U V −1 . Then by Lemma 5.7, Z := M̃ V − Ñ U is invertible in RH∞ . Define Qr by the equation U0 + M Qr = U Z −1 , (11.9) so Qr = M −1 (U Z −1 − U0 ). Then, using the Bezout identity, we have V0 + N Qr = = = = = = V0 + N M −1 (U Z −1 − U0 ) V0 + M̃ −1 Ñ(U Z −1 − U0 ) M̃ −1 (M̃ V0 − Ñ U0 + ÑU Z −1 ) M̃ −1 (I + Ñ U Z −1 ) M̃ −1 (Z + Ñ U )Z −1 M̃ −1 M̃ V Z −1 = V Z −1 . Thus, K = U V −1 = (U0 + M Qr )(V0 + N Qr )−1 . (11.10) 230 CONTROLLER PARAMETERIZATION To see that Qr belongs to RH∞ , observe first from equation (11.9) and then from equation (11.10) that both M Qr and N Qr belong to RH∞ . Then Qr = (Ṽ0 M − Ũ0 N )Qr = Ṽ0 (M Qr ) − Ũ0 (N Qr ) ∈ RH∞ . Finally, since V and Z evaluated at s = ∞ are both invertible, so is V0 + N Qr from equation (11.10), and hence so is I + V0−1 N Qr . Similarly, the parameterization given in equation (11.6) can be obtained. To show that the controller can be written in the form of equation (11.7), note that (U0 + M Qy )(V0 + N Qy )−1 = U0 V0−1 + (M − U0 V0−1 N )Qy (I + V0−1 N Qy )−1 V0−1 and that U0 V0−1 = Ṽ0−1 Ũ0 . We have (M − U0 V0−1 N ) = (M − Ṽ0−1 Ũ0 N ) = Ṽ0−1 (Ṽ0 M − Ũ0 N ) = Ṽ0−1 and K = U0 V0−1 + Ṽ0−1 Qy (I + V0−1 N Qy )−1 V0−1 . ✷ Corollary 11.7 Given an admissible controller K with coprime factorizations K = U V −1 = Ṽ −1 Ũ , the free parameter Qy ∈ RH∞ in Youla parameterization is given by Qy = M −1 (U Z −1 − U0 ) where Z := M̃V − Ñ U . Next, we shall establish the precise relationship between the preceding all stabilizing controller parameterization and the state-space parameterization in the last section. The following theorem follows from some algebraic manipulation. Theorem 11.8 Let the doubly coprime factorizations of G22 be chosen as   " # A + B2 F B2 −L M U0   = F I 0  N V0 C2 + D22 F D22 I   " # A + LC2 −(B2 + LD22 ) L Ṽ0 −Ũ0   = F I 0  −Ñ M̃ −D22 I C2 where F and L are chosen such that A + B2 F and A + LC2 are both stable. Then Jy can be computed as   A + B2 F + LC2 + LD22 F −L B2 + LD22   Jy =  F 0 I . −(C2 + D22 F ) I −D22 11.4. Notes and References 231 Remark 11.3 Note that Jy is exactly the same as the J in Theorem 11.4 and that K0 := U0 V0−1 is an observer-based stabilizing controller with # " A + B2 F + LC2 + LD22 F −L . K0 := F 0 ✸ 11.4 Notes and References The conventional Youla parameterization can be found in Youla et al. [1976a, 1976b], Desoer et al. [1980], Doyle [1984], Vidyasagar [1985], and Francis [1987]. The statespace derivation of all stabilizing controllers was reported in Lu, Zhou, and Doyle [1996]. The paper by Moore et al. [1990] contains some other related interesting results. The parameterization of all two-degree-of-freedom stabilizing controllers is given in Youla and Bongiorno [1985] and Vidyasagar [1985]. 11.5 Problems 1 . Find the set of all stabilizing controllers K = Fℓ (J, Q). s−1 Now verify that K0 = −4 is a stabilizing controller and find a Q0 ∈ RH∞ such that K0 = Fℓ (J, Q0 ). Problem 11.1 Let P = Problem 11.2 Suppose that {Pi : i = 1, . . . , n} is a set of MIMO plants and that there is a single controller K that internally stabilizes each Pi in the set. Show that there exists a single transfer function P such that the set P = {Fu (P, ∆) | ∆ ∈ H∞ , k∆k∞ ≤ 1 } is also robustly stabilized by K and that {Pi } ⊂ P. Problem 11.3 Internal Model Control (IMC): Suppose a plant P is stable. Then it is known that all stabilizing controllers can be parameterized as K(s) = Q(I − P Q)−1 for all stable Q. In practice, the exact plant model is not known, only a nominal model P0 is available. Hence the controller can be implemented as in the following diagram: r ✲e ✲e ✲ Q −✻ ✻ P0 ✛ ✲ P y ✲ 232 CONTROLLER PARAMETERIZATION The control diagram can be redrawn as follows: r ✲e −✻ ✲ Q y ✲ ✲ P ✲ P0 ❄ ✲e − This control implementation is known as internal model control (IMC). Note that no signal is fed back if the model is exact. Discuss the advantage of this implementation and possible generalizations. Problem 11.4 Use the Youla parameterization (the coprime factor form) to show that a SISO plant cannot be stabilized by a stable controller if the plant does not satisfy the parity interlacing properties. [A SISO plant is said to satisfy the parity interlacing property if the number of unstable real poles between any two unstable real zeros is even; +∞ counts as a unstable zero if the plant is strictly proper. See Youla, Jabr, and Lu [1974] and Vidyasagar [1985].] Chapter 12 Algebraic Riccati Equations We studied the Lyapunov equation in Chapter 7 and saw the roles it played in some applications. A more general equation than the Lyapunov equation in control theory is the so-called algebraic Riccati equation or ARE for short. Roughly speaking, Lyapunov equations are most useful in system analysis while AREs are most useful in control system synthesis; particularly, they play central roles in H2 and H∞ optimal control. Let A, Q, and R be real n × n matrices with Q and R symmetric. Then an algebraic Riccati equation is the following matrix equation: A∗ X + XA + XRX + Q = 0. Associated with this Riccati equation is a 2n × 2n matrix: " # A R H := . −Q −A∗ (12.1) (12.2) A matrix of this form is called a Hamiltonian matrix. The matrix H in equation (12.2) will be used to obtain the solutions to the equation (12.1). It is useful to note that the spectrum of H is symmetric about the imaginary axis. To see that, introduce the 2n × 2n matrix: " # 0 −I J := I 0 having the property J 2 = −I. Then J −1 HJ = −JHJ = −H ∗ so H and −H ∗ are similar. Thus λ is an eigenvalue iff −λ̄ is. This chapter is devoted to the study of this algebraic Riccati equation. 233 234 12.1 ALGEBRAIC RICCATI EQUATIONS Stabilizing Solution and Riccati Operator Assume that H has no eigenvalues on the imaginary axis. Then it must have n eigenvalues in Re s < 0 and n in Re s > 0. Consider the n-dimensional invariant spectral subspace, X− (H), corresponding to eigenvalues of H in Re s < 0. By finding a basis for X− (H), stacking the basis vectors up to form a matrix, and partitioning the matrix, we get # " X1 X− (H) = Im X2 where X1 , X2 ∈ Cn×n . (X1 and X2 can be chosen to be real matrices.) If X1 is nonsingular or, equivalently, if the two subspaces " # 0 X− (H), Im (12.3) I are complementary, we can set X := X2 X1−1 . Then X is uniquely determined by H (i.e., H 7−→ X is a function, which will be denoted Ric). We will take the domain of Ric, denoted dom(Ric), to consist of Hamiltonian matrices H with two properties: H has no eigenvalues on the imaginary axis and the two subspaces in equation(12.3) are complementary. For ease of reference, these will be called the stability property and the complementarity property, respectively. This solution will be called the stabilizing solution. Thus, X = Ric(H) and Ric : dom(Ric) ⊂ R2n×2n 7−→ Rn×n . Remark 12.1 It is now clear that to obtain the stabilizing solution to the Riccati equation, it is necessary to construct bases for the stable invariant subspace of H. One way of constructing this invariant subspace is to use eigenvectors and generalized eigenvectors of H. Suppose λi is an eigenvalue of H with multiplicity k (then λi+j = λi for all j = 1, . . . , k − 1), and let vi be a corresponding eigenvector and vi+1 , . . . , vi+k−1 be the corresponding generalized eigenvectors associated with vi and λi . Then vj are related by (H − λi I)vi = 0 (H − λi I)vi+1 = vi .. . (H − λi I)vi+k−1 = vi+k−2 , and the span{vj , j = i, . . . , i + k − 1} is an invariant subspace of H. The sum of all invariant subspaces corresponding to stable eigenvalues is the stable invariant subspace X− (H). ✸ 12.1. Stabilizing Solution and Riccati Operator 235 Example 12.1 Let A= " −3 2 −2 1 # R= " 0 0 0 −1 # , Q= " 0 0 0 0 # . The eigenvalues of H are 1, 1, −1, −1, and the corresponding eigenvectors and generalized eigenvectors are         1 −1 1 1          2   −3/2   1   3/2         . v1 =  v3 =   , v4 =   , v2 =   1   2     0   0  −2 0 0 0 It is easy to check that {v3 , v4 } form a basis for the stable invariant subspace X− (H), {v1 , v2 } form a basis for the antistable invariant subspace, and {v1 , v3 } form a basis for another invariant subspace corresponding to eigenvalues {1, −1} so " # " # −10 6 −2 2 X = 0, X̃ = , X̂ = 6 −4 2 −2 are all solutions of the ARE with the property λ(A + RX) = {−1, −1}, λ(A + RX̃) = {1, 1}, λ(A + RX̂) = {1, −1}. Thus only X is the stabilizing solution. The stabilizing solution can be found using the following Matlab command: ≫ [X1 , X2 ] = ric schr(H), X = X2 /X1 The following well-known results give some properties of X as well as verifiable conditions under which H belongs to dom(Ric). Theorem 12.1 Suppose H ∈ dom(Ric) and X = Ric(H). Then (i) X is real symmetric; (ii) X satisfies the algebraic Riccati equation A∗ X + XA + XRX + Q = 0; (iii) A + RX is stable. 236 ALGEBRAIC RICCATI EQUATIONS Proof. (i) Let X1 , X2 be as before. It is claimed that X1∗ X2 is Hermitian. (12.4) To prove this, note that there exists a stable matrix H− in Rn×n such that " # " # X1 X1 H = H− . X2 X2 (H− is a matrix representation of H|X− (H) .) Premultiply this equation by " to get " X1 X2 #∗ JH " X2 # X1 X2 #∗ X1 = " J #∗ X1 X2 J " X1 X2 # H− . (12.5) Since JH is symmetric, so is the left-hand side of equation (12.5) and so is the right-hand side: ∗ (−X1∗ X2 + X2∗ X1 )H− = H− (−X1∗ X2 + X2∗ X1 )∗ ∗ = −H− (−X1∗ X2 + X2∗ X1 ). This is a Lyapunov equation. Since H− is stable, the unique solution is −X1∗ X2 + X2∗ X1 = 0. This proves equation (12.4). Hence X := X2 X1−1 = (X1−1 )∗ (X1∗ X2 )X1−1 is Hermitian. Since X1 and X2 can always be chosen to be real and X is unique, X is real symmetric. (ii) Start with the equation H and postmultiply by X1−1 to get " H Now pre-multiply by [X " I X X1 X2 # # " = = " X1 X2 I X # " I X # H− X1 H− X1−1 . − I]: [X − I]H # = 0. (12.6) 12.1. Stabilizing Solution and Riccati Operator 237 This is precisely the Riccati equation. (iii) Premultiply equation (12.6) by [I 0] to get A + RX = X1 H− X1−1 . Thus A + RX is stable because H− is. ✷ Now we are going to state one of the main theorems of this section; it gives the necessary and sufficient conditions for the existence of a unique stabilizing solution of equation (12.1) under certain restrictions on the matrix R. Theorem 12.2 Suppose H has no imaginary eigenvalues and R is either positive semidefinite or negative semidefinite. Then H ∈ dom(Ric) if and only if (A, R) is stabilizable. Proof. (⇐) To prove that H ∈ dom(Ric), we must show that " # 0 X− (H), Im I are complementary. This requires a preliminary step. As in the proof of Theorem 12.1 define X1 , X2 , H− so that " # X1 X− (H) = Im X2 " # " # X1 X1 H = H− . (12.7) X2 X2 We want to show that X1 is nonsingular (i.e., Ker X1 = 0). First, it is claimed that Ker X1 is H− invariant. To prove this, let x ∈ Ker X1 . Premultiply equation (12.7) by [I 0] to get AX1 + RX2 = X1 H− . (12.8) Premultiply by x∗ X2∗ , postmultiply by x, and use the fact that X2∗ X1 is symmetric [see equation (12.4)] to get x∗ X2∗ RX2 x = 0. Since R is semidefinite, this implies that RX2 x = 0. Now postmultiply equation (12.8) by x to get X1 H− x = 0 (i.e., H− x ∈ Ker X1 ). This proves the claim. Now to prove that X1 is nonsingular, suppose, on the contrary, that Ker X1 6= 0. Then H− |Ker X1 has an eigenvalue, λ, and a corresponding eigenvector, x: H− x = λx Re λ < 0, 0 6= x ∈ Ker X1 . (12.9) 238 ALGEBRAIC RICCATI EQUATIONS Premultiply equation (12.7) by [0 I]: −QX1 − A∗ X2 = X2 H− . (12.10) Postmultiply the above equation by x and use equation (12.9): (A∗ + λI)X2 x = 0. Recall that RX2 x = 0; we have x∗ X2∗ [A + λI R] = 0. Then the stabilizability " #of (A, R) implies X2 x = 0. But if both X1 x = 0 and X2 x = 0, X1 then x = 0 since has full column rank, which is a contradiction. X2 (⇒) This is obvious since H ∈ dom(Ric) implies that X is a stabilizing solution and that A + RX is asymptotically stable. It also implies that (A, R) must be stabilizable. ✷ The following result is the so-called bounded real lemma, which follows immediately from the preceding theorem. " # A B ∈ RH∞ , and Corollary 12.3 Let γ > 0, G(s) = C D # " A + BR−1 D∗ C BR−1 B ∗ H := −C ∗ (I + DR−1 D∗ )C −(A + BR−1 D∗ C)∗ where R = γ 2 I − D∗ D. Then the following conditions are equivalent: (i) kGk∞ < γ. (ii) σ̄(D) < γ and H has no eigenvalues on the imaginary axis. (iii) σ̄(D) < γ and H ∈ dom(Ric). (iv) σ̄(D) < γ and H ∈ dom(Ric) and Ric(H) ≥ 0 (Ric(H) > 0 if (C, A) is observable). (v) σ̄(D) < γ and there exists an X = X ∗ ≥ 0 such that X(A+BR−1 D∗ C)+(A+BR−1 D∗ C)∗ X +XBR−1 B ∗ X +C ∗ (I +DR−1 D∗ )C = 0 and A + BR−1 D∗ C + BR−1 B ∗ X has no eigenvalues on the imaginary axis. (vi) σ̄(D) < γ and there exists an X = X ∗ > 0 such that X(A+BR−1 D∗ C)+(A+BR−1 D∗ C)∗ X +XBR−1 B ∗ X +C ∗ (I +DR−1 D∗ )C < 0. 12.1. Stabilizing Solution and Riccati Operator (vii) There exists an X = X ∗ > 0 such that  XA + A∗ X XB  B∗X −γI  C D 239  C∗  D∗  < 0. −γI Proof. The equivalence between (i) and (ii) has been shown in Chapter 4. The equivalence between (iii) and (iv) is obvious by noting the fact that A + BR−1 D∗ C is stable if kGk∞ < γ (See Problem 12.15). The equivalence between (ii) and (iii) follows from the preceding theorem. It is also obvious that (iv) implies (v). We shall now show that (v) implies (i). Thus suppose that there is an X ≥ 0 such that X(A + BR−1 D∗ C) + (A + BR−1 D∗ C)∗ X + XBR−1 B ∗ X + C ∗ (I + DR−1 D∗ )C = 0 and A + BR−1 (B ∗ X + D∗ C) has no eigenvalues on the imaginary axis. Then # " −B A W (s) := B ∗ X + D∗ C R has no zeros on the imaginary axis since " A + BR−1 (B ∗ X + D∗ C) −1 W (s) = R−1 (B ∗ X + D∗ C) BR−1 R−1 # has no poles on the imaginary axis. Next, note that −X(jωI − A) − (jωI − A)∗ X + XBR−1 D∗ C + C ∗ DR−1 B ∗ X +XBR−1 B ∗ X + C ∗ (I + DR−1 D∗ )C = 0. Multiplying B ∗ {(jωI−A)∗ }−1 on the left and (jωI−A)−1 B on the right of the preceding equation and completing square, we have G∗ (jω)G(jω) = γ 2 I − W ∗ (jω)R−1 W (jω). Since W (s) has no zeros on the imaginary axis, we conclude that kGk∞ < γ. The equivalence between (vi) and (vii) follows from Schur complement. It is also easy to show that (vi) implies (i) by following the similar procedure as above. To show that (i) implies (vi), let   A B   Ĝ =  C D  . ǫI Then there exists an ǫ > 0 such that Ĝ to Ĝ. ∞ 0 < γ. Now (vi) follows by applying part (v) ✷ 240 ALGEBRAIC RICCATI EQUATIONS Theorem 12.4 Suppose H has the form " A H= −C ∗ C −BB ∗ −A∗ # . Then H ∈ dom(Ric) iff (A, B) is stabilizable and (C, A) has no unobservable modes on the imaginary axis. Furthermore, X = Ric(H) ≥ 0 if H ∈ dom(Ric), and Ker(X) = {0} if and only if (C, A) has no stable unobservable modes. Note that Ker(X) ⊂ Ker(C), so that the equation XM = C ∗ always has a solution for M , and a minimum F -norm solution is given by X + C ∗ . Proof. It is clear from Theorem 12.2 that the stabilizability of (A, B) is necessary, and it is also sufficient if H has no eigenvalues on the imaginary axis. So we only need to show that, assuming (A, B) is stabilizable, H has no imaginary eigenvalues iff (C, A) has no unobservable modes on the imaginary axis. Suppose that jω is an eigenvalue " # x and 0 6= is a corresponding eigenvector. Then z Ax − BB ∗ z = jωx −C ∗ Cx − A∗ z = jωz. Rearrange: (A − jωI)x = BB ∗ z (12.11) −(A − jωI)∗ z = C ∗ Cx. (12.12) Thus hz, (A − jωI)xi = hz, BB ∗ zi = kB ∗ zk2 −hx, (A − jωI)∗ zi = hx, C ∗ Cxi = kCxk2 so hx, (A − jωI)∗ zi is real and −kCxk2 = h(A − jωI)x, zi = hz, (A − jωI)xi = kB ∗ zk2 . Therefore, B ∗ z = 0 and Cx = 0. So from equations (12.11) and (12.12) (A − jωI)x = 0 (A − jωI)∗ z = 0. Combine the last four equations to get z ∗ [A − jωI B] = 0 12.1. Stabilizing Solution and Riccati Operator " A − jωI C # 241 x = 0. The stabilizability of (A, B) gives z = 0. Now it is clear that jω is an eigenvalue of H iff jω is an unobservable mode of (C, A). Next, set X := Ric(H). We will show that X ≥ 0. The Riccati equation is A∗ X + XA − XBB ∗ X + C ∗ C = 0 or, equivalently, (A − BB ∗ X)∗ X + X(A − BB ∗ X) + XBB ∗ X + C ∗ C = 0. Noting that A − BB ∗ X is stable (Theorem 12.1), we have Z ∞ ∗ ∗ ∗ X= e(A−BB X) t (XBB ∗ X + C ∗ C)e(A−BB X)t dt. (12.13) (12.14) 0 Since XBB ∗ X + C ∗ C is positive semidefinite, so is X. Finally, we will show that KerX is nontrivial if and only if (C, A) has stable unobservable modes. Let x ∈ KerX, then Xx = 0. Premultiply equation (12.13) by x∗ and postmultiply by x to get Cx = 0. Now postmultiply equation (12.13) again by x to get XAx = 0. We conclude that Ker(X) is an A-invariant subspace. Thus if Ker(X) 6= {0}, then there is a 0 6= x ∈ Ker(X) and a λ such that λx = Ax = (A − BB ∗ X)x and Cx = 0. Since (A − BB ∗ X) is stable, Reλ < 0; thus λ is a stable unobservable mode. Conversely, suppose (C, A) has an unobservable stable mode λ (i.e., there is an x such that Ax = λx, Cx = 0). By premultiplying the Riccati equation by x∗ and postmultiplying by x, we get 2Reλx∗ Xx − x∗ XBB ∗ Xx = 0. Hence x∗ Xx = 0 (i.e., X is singular) since Reλ < 0. ✷ Example 12.2 This example shows that the observability of (C, A) is not necessary for the existence of a positive definite stabilizing solution. Let " # " # h i 1 0 1 A= , B= , C= 0 0 . 0 2 1 242 ALGEBRAIC RICCATI EQUATIONS Then (A, B) is stabilizable, but (C, A) is not detectable. However, " # 18 −24 X= >0 −24 36 is the stabilizing solution. Corollary 12.5 Suppose that (A, B) is stabilizable and (C, A) is detectable. Then the Riccati equation A∗ X + XA − XBB ∗ X + C ∗ C = 0 has a unique positive semidefinite solution. Moreover, the solution is stabilizing. Proof. It is obvious from the preceding theorem that the Riccati equation has a unique stabilizing solution and that the solution is positive semidefinite. Hence we only need to show that any positive semidefinite solution X ≥ 0 must also be stabilizing. Then by the uniqueness of the stabilizing solution, we can conclude that there is only one positive semidefinite solution. To achieve that goal, let us assume that X ≥ 0 satisfies the Riccati equation but that it is not stabilizing. First rewrite the Riccati equation as (A − BB ∗ X)∗ X + X(A − BB ∗ X) + XBB ∗ X + C ∗ C = 0 (12.15) and let λ and x be an unstable eigenvalue and the corresponding eigenvector of A − BB ∗ X, respectively; that is, (A − BB ∗ X)x = λx. Now premultiply and postmultiply equation (12.15) by x∗ and x, respectively, and we have (λ̄ + λ)x∗ Xx + x∗ (XBB ∗ X + C ∗ C)x = 0. This implies B ∗ Xx = 0, Cx = 0 since Re(λ) ≥ 0 and X ≥ 0. Finally, we arrive at Ax = λx, Cx = 0. That is, (C, A) is not detectable, which is a contradiction. Hence Re(λ) < 0 (i.e., X ≥ 0 is the stabilizing solution). ✷ Lemma 12.6 Suppose D has full column rank and let R = D∗ D > 0; then the following statements are equivalent: " # A − jωI B (i) has full column rank for all ω. C D
(ii) (I − DR−1 D∗ )C, A − BR−1 D∗ C has no unobservable modes on the jω axis. 12.1. Stabilizing Solution and Riccati Operator 243
Proof. Suppose jω is an unobservable mode of (I − DR−1 D∗ )C, A − BR−1 D∗ C ; then there is an x 6= 0 such that (A − BR−1 D∗ C)x = jωx, that is, " A − jωI C #" B D But this implies that " (I − DR−1 D∗ )Cx = 0; I 0 I −R−1 D∗ C A − jωI C B D #" # x 0 = 0. # (12.16) does not have full-column rank. Conversely, suppose " # equation (12.16) does not have u full-column rank for some ω; then there exists 6= 0 such that v " #" # A − jωI B u = 0. C D v Now let " Then " u v x y # # = = " " I −R−1 D∗ C I −1 ∗ R D C 0 I 0 I #" #" u v x y # # 6= 0 . and (A − BR−1 D∗ C − jωI)x + By = 0 (I − DR −1 (12.17) ∗ D )Cx + Dy = 0. (12.18) Premultiply equation (12.18) by D∗ to get y = 0. Then we have (A − BR−1 D∗ C)x = jωx, (I − DR−1 D∗ )Cx = 0;
that is, jω is an unobservable mode of (I − DR−1 D∗ )C, A − BR−1 D∗ C . Remark 12.2 If D is not square, then there is a D⊥ such that ∗ that D⊥ D⊥ −1 ∗ h D⊥ DR−1/2 ✷ i is unitary and = I−DR D . Hence, in some cases we will write the condition ∗ (ii) in the preceding lemma as (D⊥ C, A−BR−1 D∗ C) having no imaginary unobservable modes. Of course, if D is square, the condition is simplified to A − BR−1 D∗ C having no imaginary eigenvalues. Note also that if D∗ C = 0, condition (ii) becomes (C, A) having no imaginary unobservable modes. ✸ 244 ALGEBRAIC RICCATI EQUATIONS Corollary 12.7 Suppose D has full column rank and denote R = D∗ D > 0. Let H have the form " # " # i h A 0 B −1 ∗ ∗ H = − R D C B −C ∗ C −A∗ −C ∗ D " # −BR−1 B ∗ A − BR−1 D∗ C = . −C ∗ (I − DR−1 D∗ )C −(A − BR−1 D∗ C)∗ " # A − jωI B Then H ∈ dom(Ric) iff (A, B) is stabilizable and has full-column rank C D for all ω. Furthermore, X = Ric(H) ≥ 0 if H ∈ dom(Ric), and Ker(X) = {0} if and ∗ only if (D⊥ C, A − BR−1 D∗ C) has no stable unobservable modes. Proof. This is the consequence of Lemma 12.6 and Theorem 12.4. ✷ Remark 12.3 It is easy to see that the detectability (observability) of ∗ (D⊥ C, A − BR−1 D∗ C) implies the detectability (observability) of (C, A); however, the converse is, in general, not true. Hence the existence of a stabilizing solution to the Riccati equation in the preceding corollary is not guaranteed by the stabilizability of (A, B) and detectability of (C, A). Furthermore, even if a stabilizing solution exists, the positive definiteness of the solution is not guaranteed by the observability of (C, A) unless D∗ C = 0. As an example, consider " # " # " # " # 0 1 0 1 0 1 A= , B= , C= , D= . 0 0 −1 0 0 0 Then (C, A) is observable, (A, B) is controllable, and " # h i 0 1 ∗ ∗ A − BD C = , D⊥ C= 0 0 . 1 0 A Riccati equation with the preceding data has a nonnegative definite stabilizing so∗ C, A − BR−1 D∗ C) has no unobservable modes on the imaginary axis. lution since (D⊥ ∗ However, the solution is not positive definite since (D⊥ C, A − BR−1 D∗ C) has a stable unobservable mode. On the other hand, if the B matrix is changed to " # 0 B= , 1 then the corresponding Riccati equation has no stabilizing solution since, in this case, (A − BD∗ C) has eigenvalues on the imaginary axis although (A, B) is controllable and (C, A) is observable. ✸ Related MATLAB Commands: ric eig, are 12.2. Inner Functions 12.2 245 Inner Functions A transfer function N is called inner if N ∈ RH∞ and N ∼ N = I and co-inner if N ∈ RH∞ and N N ∼ = I. Note that N need not be square. Inner and co-inner are dual notions (i.e., N is an inner iff N T is a co-inner). A matrix function N ∈ RL∞ is called all-pass if N is square and N ∼ N = I; clearly a square inner function is all-pass. We will focus on the characterizations of inner functions here, and the properties of co-inner functions follow by duality. Note that N inner implies that N has at least as many rows as columns. For N inner and any q ∈ Cm , v ∈ L2 , kN (jω)qk = kqk, ∀ω and kN vk2 = kvk2 since N (jω)∗ N (jω) = I for all ω. Because of these norm preserving properties, inner matrices will play an important role in the control synthesis theory in this book. In this section, we present a state-space characterization of inner transfer functions. # " A B ∈ RH∞ and X = X ∗ ≥ 0 satisfies Lemma 12.8 Suppose N = C D A∗ X + XA + C ∗ C = 0. (12.19) Then (a) D∗ C + B ∗ X = 0 implies N ∼ N = D∗ D. (b) (A, B) is controllable, and N ∼ N = D∗ D implies that D∗ C + B ∗ X = 0. Proof. Conjugating the states of  by " I −X 0 I # A  ∼  N N =  −C ∗ C D∗ C on the left and N ∼N " I −X 0 I 0 −A∗ B∗ #−1 = "  B  −C ∗ D   D∗ D I X 0 I # on the right yields  B A 0   ∗ ∗ ∗ ∗  =   −(A X + XA + C C) −A −(XB + C D)  B ∗ X + D∗ C B∗ D∗ D   B A 0   0 −A∗ −(XB + C ∗ D)  =  .   B ∗ X + D∗ C Then (a) and (b) follow easily. B∗ D∗ D ✷ 246 ALGEBRAIC RICCATI EQUATIONS This lemma immediately leads to one characterization of inner matrices in terms of their state-space representations. Simply add the condition that D∗ D = I to Lemma 12.8 to get N ∼ N = I. " A B # is stable and minimal, and X is the observC D ability Gramian. Then N is an inner if and only if Corollary 12.9 Suppose N = (a) D∗ C + B ∗ X = 0 (b) D∗ D = I. A transfer matrix N⊥ is called a complementary inner factor (CIF) of N if [N N⊥ ] is square and is an inner. The dual notion of the complementary co-inner factor is defined in the obvious way. Given an inner N , the following lemma gives a construction of its CIF. The proof of this lemma follows from straightforward calculation and from the fact that CX + X = C since Im(I − X + X) ⊂ Ker(X) ⊂ Ker(C). " # A B Lemma 12.10 Let N = be an inner and X be the observability Gramian. C D Then a CIF N⊥ is given by " # A −X + C ∗ D⊥ N⊥ = C D⊥ where D⊥ is an orthogonal complement of D such that [D D⊥ ] is square and orthogonal. 12.3 Notes and References The general solutions of a Riccati equation are given by Martensson [1971]. The paper by Willems [1971] contains a comprehensive treatment of ARE and the related optimization problems. Some matrix factorization results are given in Doyle [1984]. Numerical methods for solving ARE can be found in Arnold and Laub [1984], Van Dooren [1981], and references therein. See Zhou, Doyle, and Glover [1996] and Lancaster and Rodman [1995] for a more extensive treatment of this subject. 12.4 Problems " A B # ∈ RL∞ is a stabilizable and deC D tectable realization and γ > kG(s)k∞ . Show that there exists a transfer matrix M ∈ Problem 12.1 Assume that G(s) := 12.4. Problems 247 RL∞ such that M ∼ M = γ 2 I − G∼ G and M −1 ∈ RH∞ . A particular realization of M is # " B A M (s) = −R1/2 F R1/2 where R F X = γ 2 I − D∗ D = R−1 (B ∗ X + D∗ C) " A + BR−1 D∗ C = Ric −C ∗ (I + DR−1 D∗ )C and X ≥ 0 if A is stable. Problem 12.2 Let G(s) = " A B C D BR−1 B ∗ −(A + BR−1 D∗ C)∗ # # be a stabilizable and detectable realization. " # A − jω B ∼ Suppose G (jω)G(jω) > 0 for all ω or has full-column rank for all ω. C D Let # " A − BR−1 D∗ C −BR−1 B ∗ X = Ric −C ∗ (I − DR−1 D∗ )C −(A − BR−1 D∗ C)∗ with R := D∗ D > 0. Show where W −1 ∈ RH∞ and W = W ∼ W = G∼ G " A B R−1/2 (D∗ C + B ∗ X) R1/2 # . Problem 12.3 A square (m × m) matrix function G(s) ∈ RH∞ is said to be positive real (PR) if G(jω) + G∗ (jω) ≥ 0 for all finite ω; and G(s) "is said to#be strictly positive A B real (SPR) if G(jω) + G∗ (jω) > 0 for all ω ∈ R. Let be a state-space C D realization of G(s) with A stable (not necessarily a minimal realization). Suppose there exist X ≥ 0, Q, and W such that XA + A∗ X = −Q∗ Q B∗X + W ∗Q = C D + D∗ = W ∗ W, Show that G(s) is positive real and G(s) + G∼ (s) = M ∼ (s)M (s) (12.20) (12.21) (12.22) 248 with M (s) = ALGEBRAIC RICCATI EQUATIONS " A B # . Furthermore, if M (jω) has full-column rank for all ω ∈ R, Q W then G(s) is strictly positive real. Problem 12.4 Suppose (A, B, C, D) is a minimal realization of G(s) with A stable and G(s) positive real. Show that there exist X ≥ 0, Q, and W such that XA + A∗ X = −Q∗ Q B∗X + W ∗Q = C D + D∗ = W ∗ W and G(s) + G∼ (s) = M ∼ (s)M (s) " # A B . Furthermore, if G(s) is strictly positive real, then M (jω) has Q W full-column rank for all ω ∈ R. " # A B Problem 12.5 Let be a state-space realization of G(s) ∈ RH∞ with A C D stable and R := D + D∗ > 0. Show that G(s) is strictly positive real if and only if there exists a stabilizing solution to the following Riccati equation: with M (s) = X(A − BR−1 C) + (A − BR−1 C)∗ X + XBR−1 B ∗ X + C ∗ R−1 C = 0. " # A B Moreover, M (s) = is minimal phase and 1 1 R− 2 (C − B ∗ X) R 2 G(s) + G∼ (s) = M ∼ (s)M (s). Problem 12.6 Assume p ≥ m. Show that there exists an rcf G = N M −1 such that N is an inner if and only if G∼ G > 0 on the jω axis, including at ∞. This factorization is unique that the realization of # " up to a#constant unitary multiple." Furthermore, assume A B A − jωI B is stabilizable and that has full column rank for all G= C D C D ω ∈ R. Then a particular realization of the desired coprime factorization is   " # A + BF BR−1/2 M   :=  F R−1/2  ∈ RH∞ N C + DF DR−1/2 where R = D∗ D > 0 12.4. Problems 249 F = −R−1 (B ∗ X + D∗ C) and X = Ric " A − BR−1 D∗ C −C ∗ (I − DR−1 D∗ )C −BR−1 B ∗ −(A − BR−1 D∗ C)∗ # ≥ 0. Moreover, a complementary inner factor can be obtained as " # A + BF −X † C ∗ D⊥ N⊥ = C + DF D⊥ if p > m. " A B # ∈ Rp (s) and (A, B) is stabilizable. Show C D that there exists a right coprime factorization G = N M −1 such that M ∈ RH∞ is an inner if and only if G has no poles on the jω axis. A particular realization is Problem 12.7 Assume that G = " M N #   :=  A + BF B F C + DF I D    ∈ RH∞ where F = −B ∗ X " # A −BB ∗ X = Ric ≥ 0. 0 −A∗ Problem 12.8 A right coprime factorization of G = N M −1 with N, M ∈ RH " ∞ is # M ∼ ∼ called a normalized right coprime factorization if M M + N N = I; that is, if N is an inner. Similarly, an lcf G = M̃ −1 Ñ is called a normalized left "coprime factorization # h i A B if M̃ Ñ is a co-inner. Let a realization of G be given by G = and define C D R = I + D∗ D > 0 and R̃ = I + DD∗ > 0. (a) Suppose (A, B) is stabilizable and (C, A) has no unobservable modes on the imaginary axis. Show that there is a normalized right coprime factorization G = N M −1 " M N #   :=  A + BF BR−1/2 F C + DF R−1/2 DR−1/2    ∈ RH∞ 250 ALGEBRAIC RICCATI EQUATIONS where F = −R−1 (B ∗ X + D∗ C) and X = Ric " A − BR−1 D∗ C −C ∗ R̃−1 C −BR−1 B ∗ −(A − BR−1 D∗ C)∗ # ≥ 0. (b) Suppose (C, A) is detectable and (A, B) has no uncontrollable modes on the imaginary axis. Show that there is a normalized left coprime factorization G = M̃ −1 Ñ # " h i L B + LD A + LC M̃ Ñ := R̃−1/2 C R̃−1/2 R̃−1/2 D where L = −(BD∗ + Y C ∗ )R̃−1 and Y = Ric " (A − BD∗ R̃−1 C)∗ −BR−1 B ∗ −C ∗ R̃−1 C −(A − BD∗ R̃−1 C) # ≥ 0. (c) Show " #that the controllability Gramian P and the observability Gramian Q of M are given by N P = (I + Y X)−1 Y, Q=X while the controllability Gramian P̃ and observability Gramian Q̃ of are given by P̃ = Y, Q̃ = (I + XY )−1 X. Problem 12.9 Let G(s) = " A B C D # h M̃ Ñ i . Find M1 and M2 such that M1−1 , M2−1 ∈ RH∞ and M1 M1∼ = I + GG∼ , M2∼ M2 = I + G∼ G. Problem 12.10 Let A ∈ Rm×m , B ∈ Rn×n , C ∈ Rm×n , and consider the Sylvester equation AX + XB = C for an unknown matrix X ∈ Rm×n . Let " # " B 0 B M= , N= C −A 0 0 −A # . 12.4. Problems 251 " # U 1. Let the columns of ∈ Cn+m×n be the eigenvectors of M associated with V the eigenvalues of B and suppose U is nonsingular. Show that X = V U −1 solves the Sylvester equation. Moreover, every solution of the Sylvester equation can be written in the above form. 2. Show that the Sylvester equation has a solution if and only if M and N are similar. (See Lancaster and Tismenetsky [1985, page 423].) Problem 12.11 Let A ∈ Rn×n . Show that Z t ∗ eA τ QeAτ dτ P (t) = 0 satisfies Ṗ (t) = A∗ P (t) + P (t)A + Q, P (0) = 0. Problem 12.12 A more general case of the above problem is when the given matrices are time varying and the initial condition is not zero. Let A(t), Q(t), P0 ∈ Rn×n . Show that Z P (t) = ΦT (t, t0 )P0 Φ(t, t0 ) + t ΦT (t, τ )Q(τ )Φ(t, τ )dτ t0 satisfies Ṗ (t) = A∗ P (t) + P (t)A + Q(t), P (t0 ) = P0 where Φ(t, τ ) is the state transition matrix for the system ẋ = A(t)x. Problem 12.13 Let A ∈ Rn×n , R = R∗ , Q = Q∗ . Define " # A R H= . −Q −A∗ Let Θ(t) = " Θ11 (t) Θ12 (t) Θ21 (t) Θ22 (t) # = eHt . Show that P (t) = (Θ21 (t) + Θ22 P0 )(Θ11 (t) + Θ12 (t)P0 )−1 is the solution to the following differential Riccati equation: −Ṗ (t) = A∗ P (t) + P (t)A + P RP + Q, P (0) = P0 . 252 ALGEBRAIC RICCATI EQUATIONS Problem 12.14 Let A ∈ Rn×n , R = R∗ , Q = Q∗ . Define " # A R . H= −Q −A∗ Let Θ(t) = Show that " Θ11 (t) Θ12 (t) Θ21 (t) Θ22 (t) # = eH(t−T ) . −1 P (t) = Θ21 (t)Θ11 (t) is the solution to the following differential Riccati equation: −Ṗ (t) = A∗ P (t) + P (t)A + P RP + Q, P (T ) = 0. # " A B ∈ RH∞ and kGk∞ < γ. Show that Problem 12.15 Suppose G(s) = C D A + BR−1 D∗ C with R = γ 2 I − D∗ D is stable. (Hint: Show A + B(I − ∆D/γ)−1 ∆C/γ is stable for all ∆ with k∆k ≤ 1.) Chapter 13 H2 Optimal Control In this chapter we treat the optimal control of linear time-invariant systems with a quadratic performance criterion. 13.1 Introduction to Regulator Problem Consider the following dynamical system: ẋ = Ax + B2 u, x(t0 ) = x0 (13.1) where x0 is given but arbitrary. Our objective is to find a control function u(t) defined on [t0 , T ], which can be a function of the state x(t), such that the state x(t) is driven to a (small) neighborhood of origin at time T . This is the so-called regulator problem. One might suggest that this regulator problem can be trivially solved for any T > t0 if the system is controllable. This is indeed the case if the controller can provide arbitrarily large amount of energy since, by the definition of controllability, one can immediately construct a control function that will drive the state to zero in an arbitrarily short time. However, this is not practical since any physical system has energy limitation (i.e., the actuator will eventually saturate). Furthermore, large control action can easily drive the system out of the region, where the given linear model is valid. Hence certain limitations have to be imposed on the control in practical engineering implementation. The constraints on control u may be measured in many different ways; for example, Z T Z T 2 kuk dt, sup kuk ; kuk dt, t0 t∈[t0 ,T ] t0 That is, in terms of L1 norm, L2 norm, and L∞ norm or, more generally, weighted L1 norm, L2 norm, and L∞ norm Z T Z T kWu uk2 dt, sup kWu uk kWu uk dt, t0 t∈[t0 ,T ] t0 253 254 H2 OPTIMAL CONTROL for some constant weighting matrix Wu . Similarly, one might also want to impose some constraints on the transient response x(t) in a similar fashion: Z T t0 kWx xk dt, Z T t0 2 kWx xk dt, sup kWx xk t∈[t0 ,T ] for some weighting matrix Wx . Hence the regulator problem can be posed as an optimal control problem with certain combined performance index on u and x. In this chapter, we shall be concerned exclusively with the L2 performance problem or quadratic performance problem. Moreover, we shall focus on the infinite time regulator problem (i.e., T → ∞) and, without loss of generality, we shall assume t0 = 0. In this case, our problem is as follows: Find a control u(t) defined on [0, ∞) such that the state x(t) is driven to the origin as t → ∞ and the following performance index is minimized: # #" #∗ " Z ∞" x(t) Q S x(t) dt (13.2) min u u(t) S∗ R u(t) 0 for some Q = Q∗ , S, and R = R∗ > 0. This problem is traditionally called a linear quadratic regulator problem or, simply, an LQR problem. Here we have assumed R > 0 to emphasize that the control energy has to be finite (i.e., u(t) ∈ L2 [0, ∞)). So L2 [0, ∞) is the space over which the integral is minimized. Moreover, it is also generally assumed that " # Q S ≥ 0. (13.3) S∗ R Since R is positive definite, it has a square root, R1/2 , which is also positive-definite. By the substitution u ← R1/2 u, we may as well assume at the start that R = I. In fact, we can even assume S = 0 by using a pre-state feedback u = −S ∗ x + v provided some care is exercised; however, this will not be assumed in the sequel. Since the matrix in equation (13.3) is positive semi-definite with R = I, it can be factored as " # " # i h Q S C1∗ = . C D 1 12 ∗ S∗ I D12 Then equation (13.2) can be rewritten as min u∈L2 [0,∞) 2 kC1 x + D12 uk2 . 13.2. Standard LQR Problem 255 In fact, the LQR problem is posed traditionally as the minimization problem: min u∈L2 [0,∞) kC1 x + D12 uk22 (13.4) subject to: ẋ = Ax + Bu, x(0) = x0 (13.5) without explicitly mentioning the condition that the control should drive the state to the origin. Instead some assumptions are imposed on Q, S, and R (or, equivalently, on C1 and D12 ) to ensure that the optimal control law u has this property. To see what assumption one needs to make to ensure that the minimization problem formulated in equations (13.4) and (13.5) has a sensible solution, let us consider a simple example with A = 1, B = 1, Q = 0, S = 0, and R = 1: Z ∞ min u2 dt, ẋ = x + u, x(0) = x0 . u∈L2 [0,∞) 0 It is clear that u = 0 is the optimal solution. However, the system with u = 0 is unstable and x(t) diverges exponentially to infinity since x(t) = et x0 . The problem with this example is that this performance index does not “see” the unstable state x. Thus, to ensure that the minimization problem in equations (13.4) and (13.5) is sensible, we must assume that all unstable states can be “seen” from the performance index; that is, (C1 , A) must be detectable. An LQR problem with such an assumption will be called a standard LQR problem. On the other hand, if the closed-loop stabilityR is imposed on the preceding mini∞ mization, then it can be shown that minu∈L2 [0,∞) 0 u2 dt = 2x20 and u(t) = −2x(t) is the optimal control. This can also be generalized to a more general case where (C1 , A) is not necessarily detectable. Such a LQR problem will be referred to as an Extended LQR problem. 13.2 Standard LQR Problem In this section, we shall consider the LQR problem as traditionally formulated. Standard LQR Problem Let a dynamical system be described by ẋ = Ax + B2 u, x(0) = x0 given but arbitrary z = C1 x + D12 u (13.6) (13.7) and suppose that the system parameter matrices satisfy the following assumptions: (A1) (A, B2 ) is stabilizable; 256 H2 OPTIMAL CONTROL (A2) D12 has full column rank with h D12 D⊥ i unitary; (A3) (C1 , A) is detectable; # " A − jωI B2 has full column rank for all ω. (A4) C1 D12 Find an optimal control law u ∈ L2 [0, ∞) such that the performance criterion 2 kzk2 is minimized. Remark 13.1 Assumption (A1) is clearly necessary for the existence of a stabilizing control function u. The assumption (A2) is made for simplicity of notation and is ∗ actually a restatement that R = D12 D12 = I. Note also that D⊥ drops out when D12 is square. It is interesting to point out that (A3) is not needed in the Extended LQR problem. The assumption (A3) enforces that the unconditional optimization problem will result in a stabilizing control law. In fact, the assumption (A3) together with (A1) guarantees that the input/output stability implies the internal stability; that is, u ∈ L2 and z ∈ L2 imply x ∈ L2 , which will be shown in Lemma 13.1. Finally note that (A4) ∗ ∗ is equivalent to the condition that (D⊥ C1 , A − B2 D12 C1 ) has no unobservable modes on the imaginary axis and is weaker than the popular assumption of detectability of ∗ ∗ C1 , A − B2 D12 C1 ). (A4), together with the stabilizability of (A, B2 ), guarantees (D⊥ by Corollary 12.7 that the following Hamiltonian matrix belongs to dom(Ric) and that X = Ric(H) ≥ 0: H = " = " A −C1∗ C1 0 −A∗ ∗ C1 A − B2 D12 ∗ ∗ −C1 D⊥ D⊥ C1 # − " B2 −C1∗ D12 # h −B2 B2∗ ∗ C1 )∗ −(A − B2 D12 ∗ C1 D12 # B2∗ i . (13.8) ∗ Note also that if D12 C1 = 0, then (A4) is implied by the detectability of (C1 , A). ✸ Note that the Riccati equation corresponding to equation (13.8) is ∗ ∗ ∗ (A − B2 D12 C1 )∗ X + X(A − B2 D12 C1 ) − XB2 B2∗ X + C1∗ D⊥ D⊥ C1 = 0. (13.9) Now let X be the corresponding stabilizing solution and define ∗ F := −(B2∗ X + D12 C1 ). Then A + B2 F is stable. Denote AF := A + B2 F, CF := C1 + D12 F (13.10) 13.2. Standard LQR Problem 257 and rearrange equation (13.9) to get A∗F X + XAF + CF∗ CF = 0. (13.11) Thus X is the observability Gramian of (CF , AF ). Consider applying the control law u = F x to the system equations (13.6) and (13.7). The controlled system becomes ẋ = AF x, z = CF x x(0) = x0 or, equivalently, ẋ = AF x + x0 δ(t), z = CF x. x(0− ) = 0 The associated transfer matrix is Gc (s) = and " AF I CF 0 # kGc x0 k22 = x∗0 Xx0 . The proof of the following theorem requires a preliminary result about internal stability given input-output stability. Lemma 13.1 If u, z ∈ L2 [0, ∞) and (C1 , A) is detectable in the system described by equations (13.6) and (13.7), then x ∈ L2 [0, ∞). Furthermore, x(t) → 0 as t → ∞. Proof. Since (C1 , A) is detectable, there exists L such that A + LC1 is stable. Let x̂ be the state estimate of x given by x̂˙ = (A + LC1 )x̂ + (LD12 + B2 )u − Lz. Then x̂ ∈ L2 [0, ∞) and x̂ → 0 (see Problem 13.1) since z and u are in L2 [0, ∞). Now let e = x − x̂; then ė = (A + LC1 )e and e ∈ L2 [0, ∞). Therefore, x = e + x̂ ∈ L2 [0, ∞). It is easy to see that e(t) → 0 as t → ∞ for any initial condition e(0). Finally, x(t) → 0 since x̂ → 0. ✷ Theorem 13.2 There exists a unique optimal control for the LQR problem, namely u = F x. Moreover, min kzk2 = kGc x0 k2 . u∈L2 [0,∞) Note that the optimal control strategy is a constant gain state feedback, and this gain is independent of the initial condition x0 . 258 H2 OPTIMAL CONTROL Proof. With the change of variable v = u − F x, the system can be written as #" # " # " x ẋ AF B2 , x(0) = x0 . (13.12) = CF D12 v z Now if v ∈ L2 [0, ∞), then x, z ∈ L2 [0, ∞) and x(∞) = 0 since AF is stable. Hence u = F x + v ∈ L2 [0, ∞). Conversely, if u, z ∈ L2 [0, ∞), then from Lemma 13.1 x ∈ L2 [0, ∞). So v ∈ L2 [0, ∞). Thus the mapping v = u − F x between v ∈ L2 [0, ∞) and those u ∈ L2 [0, ∞) that make z ∈ L2 [0, ∞) is one-to-one and onto. Therefore, min u∈L2 [0,∞) kzk2 = min v∈L2 [0,∞) kzk2 . By differentiating x(t)∗ Xx(t) with respect to t along a solution of the differential equation (13.12) and by using equation (13.9) and the fact that CF∗ D12 = −XB2 , we see that d ∗ x Xx = ẋ∗ Xx + x∗ X ẋ = x∗ (A∗F X + XAF )x + 2x∗ XB2 v dt = −x∗ CF∗ CF x + 2x∗ XB2 v = −(CF x + D12 v)∗ (CF x + D12 v) + 2x∗ CF∗ D12 v + v ∗ v + 2x∗ XB2 v = − kzk2 + kvk2 . (13.13) Now integrate equation (13.13) from 0 to ∞ to get 2 2 kzk2 = x∗0 Xx0 + kvk2 . Clearly, the unique optimal control is v = 0, i.e., u = F x. 13.3 ✷ Extended LQR Problem This section considers the extended LQR problem where no detectability assumption is made for (C1 , A). Extended LQR Problem Let a dynamical system be given by ẋ = Ax + B2 u, x(0) = x0 given but arbitrary z = C1 x + D12 u with the following assumptions: (A1) (A, B2 ) is stabilizable; 13.4. Guaranteed Stability Margins of LQR 259 i h (A2) D12 has full column rank with D12 D⊥ unitary; " # A − jωI B2 (A3) has full column rank for all ω. C1 D12 Find an optimal control law u ∈ L2 [0, ∞) such that the system is internally stable (i.e., x ∈ L2 [0, ∞)) and the performance criterion kzk22 is minimized. Assume the same notation as in the last section, we have: Theorem 13.3 There exists a unique optimal control for the extended LQR problem, namely u = F x. Moreover, min u∈L2 [0,∞) kzk2 = kGc x0 k2 . Proof. The proof of this theorem is very similar to the proof of the standard LQR problem except that, in this case, the input/output stability may not necessarily imply the internal stability. Instead, the internal stability is guaranteed by the way of choosing control law. Suppose that u ∈ L2 [0, ∞) is such a control law that the system is stable, i.e., x ∈ L2 [0, ∞). Then v = u − F x ∈ L2 [0, ∞). On the other hand, let v ∈ L2 [0, ∞) and consider " # " #" # ẋ AF B2 x = , x(0) = x0 . z CF D12 v Then x, z ∈ L2 [0, ∞) and x(∞) = 0 since AF is stable. Hence u = F x + v ∈ L2 [0, ∞). Again the mapping v = u − F x between v ∈ L2 [0, ∞) and those u ∈ L2 [0, ∞) that make z ∈ L2 [0, ∞) and x ∈ L2 [0, ∞) is one to one and onto. Therefore, min u∈L2 [0,∞) kzk2 = min v∈L2 [0,∞) kzk2 . Using the same technique as in the proof of the standard LQR problem, we have kzk22 = x∗0 Xx0 + kvk22 . Thus, the unique optimal control is v = 0, i.e., u = F x. 13.4 ✷ Guaranteed Stability Margins of LQR Now we will consider the system described by equation (13.6) with the LQR control law u = F x. The closed-loop block diagram is as shown in Figure 13.1. The following result is the key to stability margins of an LQR control law. ∗ C1 ) and define G12 = D12 + C1 (sI − A)−1 B2 . Lemma 13.4 Let F = −(B2∗ X + D12 Then

I − B2∗ (−sI − A∗ )−1 F ∗ I − F (sI − A)−1 B2 = G∼ 12 (s)G12 (s). 260 H2 OPTIMAL CONTROL ✲ F u ✲ ẋ = Ax + B2 u x ✲ Figure 13.1: LQR closed-loop system Proof. Note that the Riccati equation (13.9) can be written as XA + A∗ X − F ∗ F + C1∗ C1 = 0. Add and subtract sX to the above equation to get −X(sI − A) − (−sI − A∗ )X − F ∗ F + C1∗ C1 = 0. Now multiply the above equation from the left by B2∗ (−sI − A∗ )−1 and from the right by (sI − A)−1 B2 to get −B2∗ (−sI − A∗ )−1 XB2 − B2∗ X(sI − A)−1 B2 − B2∗ (−sI − A∗ )−1 F ∗ F (sI − A)−1 B2 +B2∗ (−sI − A∗ )−1 C1∗ C1 (sI − A)−1 B2 = 0. ∗ Using −B2∗ X = F + D12 C1 in the above equation, we have B2∗ (−sI − A∗ )−1 F ∗ + F (sI − A)−1 B2 − B2∗ (−sI − A∗ )−1 F ∗ F (sI − A)−1 B2 ∗ +B2∗ (−sI − A∗ )−1 C1∗ D12 + D12 C1 (sI − A)−1 B2 +B2∗ (−sI − A∗ )−1 C1∗ C1 (sI − A)−1 B2 = 0. ∗ Then the result follows from completing the square and from the fact that D12 D12 = I. ✷ ∗ Corollary 13.5 Suppose D12 C1 = 0. Then

I − B2∗ (−sI − A∗ )−1 F ∗ I − F (sI − A)−1 B2 = I+B2∗ (−sI−A∗ )−1 C1∗ C1 (sI−A)−1 B2 . In particular, I − B2∗ (−jωI − A∗ )−1 F ∗ and I + B2∗ (−jωI − A∗ − F ∗ B2∗ )−1 F ∗


I − F (jωI − A)−1 B2 ≥ I
I + F (jωI − A − B2 F )−1 B2 ≤ I. (13.14) (13.15) 13.5. Standard H2 Problem 261 Note that the inequality (13.15) follows from taking the inverse of inequality (13.14). Define G(s) = −F (sI − A)−1 B2 and assume for the moment that the system is single-input. Then the inequality (13.14) shows that the open-loop Nyquist diagram of the system G(s) in Figure 13.1 never enters the unit disk centered at (−1, 0) of the complex plane. Hence the system has at least a 6 dB (= 20 log 2) gain margin and a 60o phase margin in both directions. A similar interpretation may be generalized to multiple-input systems. Next, it is noted that the inequality (13.15) can also be given some robustness interpretation. In fact, it implies that the closed-loop system in Figure 13.1 is stable even if the open-loop system G(s) is perturbed additively by a ∆ ∈ RH∞ as long as k∆k∞ < 1. This can be seen from the following block diagram and the small gain theorem, where the transfer matrix from w to z is exactly I + F (jωI − A − B2 F )−1 B2 . z ✲ F 13.5 w ✲ ∆ ✲ ẋ = Ax + B2 u ✲❄ e ✲ Standard H2 Problem The system considered in this section is described by the following standard block diagram: ✛z ✛w G y ✲ K ✛ u The realization of the transfer matrix G is taken to be of the form   B2 A B1   G(s) =  C1 0 D12  . 0 C2 D21 Notice the special off-diagonal structure of D: D22 is assumed to be zero so that G22 is strictly proper;1 also, D11 is assumed to be zero in order to guarantee that the H2 1 This assumption is made without loss of generality since a substitution of K = K(I + D K)−1 22 D would give the controller for D22 6= 0. 262 H2 OPTIMAL CONTROL problem is properly posed.2 The following additional assumptions are made for the output feedback H2 problem in this chapter: (i) (A, B2 ) is stabilizable and (C2 , A) is detectable; ∗ ∗ (ii) R1 = D12 D12 > 0 and R2 = D21 D21 > 0; " # A − jωI B2 (iii) has full column rank for all ω; C1 D12 # " A − jωI B1 has full row rank for all ω. (iv) C2 D21 The first assumption is for the stabilizability of G by output feedback, and the third and the fourth assumptions together with the first guarantee that the two Hamiltonian matrices associated with the following H2 problem belong to dom(Ric). The assumptions in (ii) guarantee that the H2 optimal control problem is nonsingular. H2 Problem The H2 control problem is to find a proper, real rational controller K that stabilizes G internally and minimizes the H2 norm of the transfer matrix Tzw from w to z. In the following discussions we shall assume that we have state models of G and K. Recall that a controller is said to be admissible if it is internally stabilizing and proper. By Corollary 12.7 the two Hamiltonian matrices " # ∗ A − B2 R1−1 D12 C1 −B2 R1−1 B2∗ H2 := ∗ ∗ C1 )∗ )C1 −(A − B2 R1−1 D12 −C1∗ (I − D12 R1−1 D12 " # ∗ (A − B1 D21 R2−1 C2 )∗ −C2∗ R2−1 C2 J2 := ∗ ∗ R2−1 C2 ) R2−1 D21 )B1∗ −(A − B1 D21 −B1 (I − D21 belong to dom(Ric), and, moreover, X2 := Ric(H2 ) ≥ 0 and Y2 := Ric(J2 ) ≥ 0. Define ∗ F2 := −R1−1 (B2∗ X2 + D12 C1 ), ∗ L2 := −(Y2 C2∗ + B1 D21 )R2−1 and AF2 := A + B2 F2 , C1F2 := C1 + D12 F2 AL2 := A + L2 C2 , B1L2 := B1 + L2 D21 Gc (s) := " Â2 := A + B2 F2 + L2 C2 # " AL2 AF2 I , Gf (s) := C1F2 0 I B1L2 0 # . Before stating the main theorem, we note the following fact: 2 Recall that a rational proper stable transfer function is an RH2 function iff it is strictly proper. 13.5. Standard H2 Problem 263 Lemma 13.6 Let U, V ∈ RH∞ be defined as " # " −1/2 B2 R1 AF2 AL2 U := , V := −1/2 −1/2 C1F2 D12 R1 R2 C2 B1L2 −1/2 R2 D21 # . ∼ Then U is an inner and V is a co-inner, U ∼ Gc ∈ RH⊥ ∈ RH⊥ 2 , and Gf V 2. Proof. The proof uses standard manipulations of state-space realizations. From U we get # " ∗ −C1F −A∗F2 ∼ 2 U (s) = . −1/2 −1/2 ∗ R1 B2∗ R1 D12 Then it is easy to compute   U ∼U =  −A∗F2 0 −1/2 R1 −1/2 B2∗   U ∼ Gc =  −1/2 ∗ −C1F D12 R1 2 −1/2 B2 R1 ∗ −C1F C1F2 2 AF2 R1 ∗ D12 C1F2 ∗ −C1F C1F2 2 AF2 −A∗F2 0 −1/2 R1 Now do the similarity transformation " −1/2 B2∗ R1 I −X2 0 I ∗ D12 C1F2    I  0  I . 0 # on the states of the preceding transfer matrices and note that ∗ C1F2 = 0. A∗F2 X2 + X2 AF2 + C1F 2 We get  ∼   U Gc =   U ∼U =  −A∗F2 0 −1/2 R1 B2∗ −A∗F2 0 0 AF2 −1/2 0  R1 0 AF2 0 B2∗  0 −1/2  B2 R1 =I I " −X2 −A∗F2  I = −1/2 R1 B2∗ 0 It follows by duality that Gf V ∼ ∈ RH⊥ 2 and V is a co-inner. −X2 0 # ∈ RH⊥ 2. ✷ 264 H2 OPTIMAL CONTROL Theorem 13.7 There exists a unique optimal controller # " Â2 −L2 . Kopt (s) := F2 0 Moreover, 1/2 min kTzw k22 = kGc B1 k22 + kR1 F2 Gf k22 = trace (B1∗ X2 B1 ) + trace (R1 F2 Y2 F2∗ ) . Proof. Consider the all-stabilizing controller parameterization K(s) = Fℓ (M2 , Q), Q ∈ RH∞ with   Â2 −L2 B2   M2 (s) =  F2 0 I  −C2 I 0 and consider the following system diagram: z✛ ✛ w G ✛ y u ❳❳❳ ✘✘✘ ✘ ❳ ✘✘✘ ❳❳❳ ✛ M2 ✛ y1 u1 ✲ Then Tzw = Fℓ (N, Q) with  AF2   0 N =  C  1F2 0 and Q −B2 F2 AL2 B1 B1L2 −D12 F2 0 D21 C2  B2  0   D12   0 1/2 1/2 1/2 Tzw = Gc B1 − U R1 F2 Gf + U R1 QR2 V. It follows from Lemma 13.6 that Gc B1 and U are orthogonal. Thus 2 kTzw k2 1/2 1/2 2 1/2 = kGc B1 k2 + U R1 F2 Gf − U R1 QR2 V = 2 kGc B1 k2 + 1/2 R1 F2 Gf − 1/2 1/2 R1 QR2 V 2 2 . 2 2 13.6. Stability Margins of H2 Controllers 265 Since Gf and V are also orthogonal by Lemma 13.6, we have kTzw k22 1/2 1/2 1/2 = kGc B1 k22 + R1 F2 Gf − R1 QR2 V 2 2 1/2 = kGc B1 k2 + R1 F2 Gf 2 1/2 2 2 1/2 + R1 QR2 2 2 . This shows clearly that Q = 0 gives the unique optimal control, so K = Fℓ (M2 , 0) is the unique optimal controller. ✷ The optimal H2 controller, Kopt , and the closed-loop transfer matrix, Tzw , can be obtained by the following Matlab program: ≫ [K, Tzw ] = h2syn(G, ny , nu ) where ny and nu are the dimensions of y and u, respectively. Related MATLAB Commands: lqg, lqr, lqr2, lqry, reg, lqe 13.6 Stability Margins of H2 Controllers It is well-known that a system with LQR controller has at least 60o phase margin and 6 dB gain margin. However, it is not clear whether these stability margins will be preserved if the states are not available and the output feedback H2 (or LQG) controller has to be used. The answer is provided here through a counterexample from Doyle [1978]: There are no guaranteed stability margins for a H2 controller. Consider a single-input and single-output two-state generalized dynamical system: # # " #  " √  " 1 1 0 σ 0 √   0 1 σ 0 1      "  " # # √ √   G(s) =  . 0 q q   0   1 0 0     h i h i 0 1 0 0 1 It can be shown analytically that # " 2α α , X2 = α α Y2 = " 2β β and F2 = −α h 1 1 i , L2 = −β " β β # 1 1 # 266 H2 OPTIMAL CONTROL where α=2+ p 4+q , β =2+ √ 4 + σ. Then the optimal output H2 controller is given by   1−β 1 β   Kopt =  −(α + β) 1 − α β  . −α −α 0 Suppose that the resulting closed-loop controller (or plant G22 ) has a scalar gain k with a nominal value k = 1. Then the controller implemented in the system is actually K = kKopt , and the closed-loop system A matrix  1   0 à =   β  β becomes 1 0 1 −kα 0 1−β 0 −α − β 0 −kα 1 1−α    .   It can be shown that the characteristic polynomial has the form det(sI − Ã) = s4 + a3 s3 + a2 s2 + a1 s + a0 with a1 = α + β − 4 + 2(k − 1)αβ, a0 = 1 + (1 − k)αβ. Note that for closed-loop stability it is necessary to have a0 > 0 and a1 > 0. Note also that a0 ≈ (1 − k)αβ and a1 ≈ 2(k − 1)αβ for sufficiently large α and β if k 6= 1. It is easy to see that for sufficiently large α and β (or q and σ), the system is unstable for arbitrarily small perturbations in k in either direction. Thus, by choice of q and σ, the gain margins may be made arbitrarily small. It is interesting to note that the margins deteriorate as control weight (1/q) gets small (large q) and/or system driving noise gets large (large σ). In modern control folklore, these have often been considered ad hoc means of improving sensitivity. It is also important to recognize that vanishing margins are not only associated with open-loop unstable systems. It is easy to construct minimum phase, open-loop stable counterexamples for which the margins are arbitrarily small. The point of this example is that H2 (LQG) solutions, unlike LQR solutions, provide no global system-independent guaranteed robustness properties. Like their more classical colleagues, modern LQG designers are obliged to test their margins for each specific design. It may, however, be possible to improve the robustness of a given design by relaxing the optimality of the filter with respect to error properties. A successful approach in 13.7. Notes and References 267 this direction is the so called LQG loop transfer recovery (LQG/LTR) design technique. The idea is to design a filtering gain, L2 , in such way so that the LQG (or H2 ) control law will approximate the loop properties of the regular LQR control. This will not be explored further here; interested readers may consult related references. 13.7 Notes and References Detailed treatment of H2 related theory, LQ optimal control, Kalman filtering, etc., can be found in Anderson and Moore [1989] or Kwakernaak and Sivan [1972]. The LQG/LTR control design was first introduced by Doyle and Stein [1981], and much work has been reported in this area since then. Additional results on the LQR stability margins can be found in Zhang and Fu [1996]. 13.8 Problems Problem 13.1 Let v(t) ∈ L2 [0, ∞). Let y(t) be the output of the system G(s) = with input v. Prove that limt→∞ y(t) = 0. 1 s+1 Problem 13.2 Parameterize all stabilizing controllers satisfying kTzw k2 ≤ γ for a given γ > 0. Problem 13.3 Consider the feedback system in Figure 6.3 and suppose P = 1 s+2 s − 10 , We = , Wu = . (s + 1)(s + 10) s + 0.001 s + 10 Design a controller that minimizes " We So Wu KSo # . 2 Simulate the time response of the system when r is a step. Problem 13.4 Repeat Problem 13.3 when We = 1/s. (Note that the solution given in this chapter cannot be applied directly.) Problem 13.5 Consider the model matching (or reference) control problem shown here: 268 H2 OPTIMAL CONTROL uw ✲ ✲ W r ✲e ✲ K − ✻ u ✲ P ✲ M ❄− e ✻ e ✲ Let M (s) ∈ H∞ be a strictly proper transfer matrix and W (s), W −1 (s) ∈ RH∞ . Formulate an H2 control problem that minimizes uw and the error e through minimizing the H2 norm of the transfer matrix from r to (e, uw ). Apply your formula to M (s) = 0.1(s + 1) 10(s + 2) 4 , W (s) = , P (s) = . s2 + 2s + 4 (s + 1)3 s + 10 Problem 13.6 Repeat Problem 13.5 with W = ǫ for ǫ = 0.01 and 0.0001. Study the behavior of the controller when ǫ → 0. Problem 13.7 Repeat Problem 13.5 and Problem 13.6 with P = 10(2 − s) . (s + 1)3 Chapter 14 H∞ Control In this chapter we consider H∞ control theory. Specifically, we formulate the optimal and suboptimal H∞ control problems in Section 14.1. However, we will focus on the suboptimal case in this book and discuss why we do so. In Section 14.2 a suboptimal controller is characterized together with an algebraic proof for a class of simplified problems while leaving the more general problems to a later section. The behavior of the H∞ controller as a function of performance level γ is considered in Section 14.3. The optimal controllers are also briefly considered in this section. Some other interpretations of the H∞ controllers are given in Section 14.4. Section 14.5 presents the formulas for an optimal H∞ controller. Section 14.6 considers again the standard H∞ control problem but with some assumptions in the previous sections relaxed. Since the proof techniques in Section 14.2 can, in principle, be applied to this general case except with some more involved algebra, the detailed proof for the general case will not be given; only the formulas are presented. We shall indicate how the assumptions in the general case can be relaxed further to accommodate other more complicated problems in Section 14.7. Section 14.8 considers the integral control in the H2 and H∞ theory and Section 14.9 considers how the general H∞ solution can be used to solve the H∞ filtering problem. 14.1 Problem Formulation Consider the system described by the block diagram ✛z ✛w G ✛ y ✲ K u where the plant G and controller K are assumed to be real rational and proper. It will be assumed that state-space models of G and K are available and that their realizations 269 270 H∞ CONTROL are assumed to be stabilizable and detectable. Recall again that a controller is said to be admissible if it internally stabilizes the system. Clearly, stability is the most basic requirement for a practical system to work. Hence any sensible controller has to be admissible. Optimal H∞ Control: Find all admissible controllers K(s) such that kTzw k∞ is minimized. It should be noted that the optimal H∞ controllers as just defined are generally not unique for MIMO systems. Furthermore, finding an optimal H∞ controller is often both numerically and theoretically complicated, as shown in Glover and Doyle [1989]. This is certainly in contrast with the standard H2 theory, in which the optimal controller is unique and can be obtained by solving two Riccati equations without iterations. Knowing the achievable optimal (minimum) H∞ norm may be useful theoretically since it sets a limit on what we can achieve. However, in practice it is often not necessary and sometimes even undesirable to design an optimal controller, and it is usually much cheaper to obtain controllers that are very close in the norm sense to the optimal ones, which will be called suboptimal controllers. A suboptimal controller may also have other nice properties (e.g., lower bandwidth) over the optimal ones. Suboptimal H∞ Control: Given γ > 0, find all admissible controllers K(s), if there are any, such that kTzw k∞ < γ. For the reasons mentioned above, we focus our attention in this book on suboptimal control. When appropriate, we briefly discuss what will happen when γ approaches the optimal value. 14.2 A Simplified H∞ Control Problem The realization of the transfer matrix G is taken to be  B2 A B1  G(s) =  C1 0 D12 0 C2 D21 The following assumptions are made: (i) (A, B1 ) is controllable and (C1 , A) is observable; (ii) (A, B2 ) is stabilizable and (C2 , A) is detectable; i h i h ∗ (iii) D12 C1 D12 = 0 I ; # " # " 0 B1 ∗ D21 = . (iv) D21 I of the form   . 14.2. A Simplified H∞ Control Problem 271 Two additional assumptions that are implicit in the assumed realization for G(s) are that D11 = 0 and D22 = 0. As we mentioned in the last chapter, D22 6= 0 does not pose any problem since it is easy to form an equivalent problem with D22 = 0 by a linear fractional transformation on the controller K(s). However, relaxing the assumption D11 = 0 complicates the formulas substantially. The H∞ solution involves the following two Hamiltonian matrices: # " # " A∗ γ −2 C1∗ C1 − C2∗ C2 A γ −2 B1 B1∗ − B2 B2∗ , J∞ := . H∞ := −C1∗ C1 −A∗ −B1 B1∗ −A The important difference here from the H2 problem is that the (1,2)-blocks are not sign definite, so we cannot use Theorem 12.4 in Chapter 12 to guarantee that H∞ ∈ dom(Ric) or Ric(H∞ ) ≥ 0. Indeed, these conditions are intimately related to the existence of H∞ suboptimal controllers. Note that the (1,2)-blocks are a suggestive combination of expressions from the H∞ norm characterization in Chapter 4 (or bounded real lemma in Chapter 12) and from the H2 synthesis of Chapter 13. It is also clear that if γ approaches infinity, then these two Hamiltonian matrices become the corresponding H2 control Hamiltonian matrices. The reasons for the form of these expressions should become clear through the discussions and proofs for the following theorem. Theorem 14.1 There exists an admissible controller such that kTzw k∞ < γ iff the following three conditions hold: (i) H∞ ∈ dom(Ric) and X∞ := Ric(H∞ ) > 0; (ii) J∞ ∈ dom(Ric) and Y∞ := Ric(J∞ ) > 0; (iii) ρ(X∞ Y∞ ) < γ 2 . Moreover, when these conditions hold, one such controller is # " Â∞ −Z∞ L∞ Ksub (s) := F∞ 0 where Â∞ := A + γ −2 B1 B1∗ X∞ + B2 F∞ + Z∞ L∞ C2 F∞ := −B2∗ X∞ , L∞ := −Y∞ C2∗ , Z∞ := (I − γ −2 Y∞ X∞ )−1 . Furthermore, the set of all admissible controllers such that kTzw k∞ < γ equals the set of all transfer matrices from y to u in ✛u ✛y   Â∞ −Z∞ L∞ Z∞ B2 M∞ ✛   M∞ (s) =  F∞ 0 I  ✲ Q where Q ∈ RH∞ , kQk∞ < γ. −C2 I 0 272 H∞ CONTROL We shall only give a proof of the first part of the theorem; the proof for the allcontroller parameterization needs much more work and is omitted (see Zhou, Doyle, and Glover [1996] for a comprehensive treatment of the related topics). We shall first show some preliminary results. Lemma 14.2 Suppose that X ∈ Rn×n , Y ∈ Rn×n , with X = X ∗ > 0, and Y = Y ∗ > 0. Let r be a positive integer. Then there exist matrices X12 ∈ Rn×r , X2 ∈ Rr×r such that X2 = X2∗ " X ∗ X12 X12 X2 # >0 " and X ∗ X12 X12 X2 #−1 = " Y ⋆ ⋆ ⋆ # if and only if " X In In Y # ≥0 and rank " X In In Y # ≤ n + r. Proof. (⇐) By the assumption, there is a matrix X12 ∈ Rn×r such that X − Y −1 = ∗ X12 X12 . Defining X2 := Ir completes the construction. (⇒) Using Schur complements, ∗ ∗ X −1 X12 )−1 X12 X −1 . Y = X −1 + X −1 X12 (X2 − X12 Inverting, using the matrix inversion lemma, gives ∗ Y −1 = X − X12 X2−1 X12 . ∗ ∗ )≤ ≥ 0, and, indeed, rank(X −Y −1 ) =rank(X12 X2−1 X12 Hence, X −Y −1 = X12 X2−1 X12 r. ✷ Lemma 14.3 There exists an rth-order admissible controller such that kTzw k∞ < γ only if the following three conditions hold: (i) There exists a Y1 > 0 such that AY1 + Y1 A∗ + Y1 C1∗ C1 Y1 /γ 2 + B1 B1∗ − γ 2 B2 B2∗ < 0. (14.1) (ii) There exists an X1 > 0 such that X1 A + A∗ X1 + X1 B1 B1∗ X1 /γ 2 + C1∗ C1 − γ 2 C2∗ C2 < 0. (iii) " X1 /γ In In Y1 /γ # ≥0 rank " X1 /γ In In Y1 /γ # ≤ n + r. (14.2) 14.2. A Simplified H∞ Control Problem 273 Proof. Suppose that there exists an rth-order controller K(s) such that kTzw k∞ < γ. Let K(s) have a state-space realization # "  B̂ . K(s) = Ĉ D̂ Then Tzw Denote  A + B2 D̂C2   B̂C2 = Fℓ (G, K) =  C1 + D12 D̂C2 R = γ 2 I − Dc∗ Dc , " By Corollary 12.3, there exists an X̃ = B2 Ĉ  D12 Ĉ  " B1 + B2 D̂D21  Ac  B̂D21  =: C c D12 D̂D21 Bc Dc # . R̃ = γ 2 I − Dc Dc∗ . # X1 X12 > 0 such that ∗ X12 X2 X̃(Ac + Bc R−1 Dc∗ Cc ) + (Ac + Bc R−1 Dc∗ Cc )∗ X̃ + X̃Bc R−1 Bc∗ X̃ + Cc∗ R̃−1 Cc < 0. (14.3) This gives, after much algebraic manipulation, X1 A + A∗ X1 + X1 B1 B1∗ X1 /γ 2 + C1∗ C1 − γ 2 C2∗ C2 +(X1 B1 D̂ + X12 B̂ + γ 2 C2∗ )(γ 2 I − D̂∗ D̂)−1 (X1 B1 D̂ + X12 B̂ + γ 2 C2∗ )∗ < 0, which implies that X1 A + A∗ X1 + X1 B1 B1∗ X1 /γ 2 + C1∗ C1 − γ 2 C2∗ C2 < 0. On the other hand, let and partition Ỹ as Ỹ = " Y1 Y12 ∗ Y12 Y2 Ỹ = γ 2 X̃ −1 # > 0. Then (Ac + Bc R−1 Dc∗ Cc )Ỹ + Ỹ (Ac + Bc R−1 Dc∗ Cc )∗ + Ỹ Cc∗ R̃−1 Cc Ỹ + Bc R−1 Bc∗ < 0. (14.4) This gives AY1 + Y1 A∗ + B1 B1∗ − γ 2 B2 B2∗ + Y1 C1∗ C1 Y1 /γ 2 +(Y1 C1∗ D̂∗ + Y12 Ĉ ∗ + γ 2 B2 )(γ 2 I − D̂D̂∗ )−1 (Y1 C1∗ D̂∗ + Y12 Ĉ ∗ + γ 2 B2 )∗ < 0, which implies that AY1 + Y1 A∗ + B1 B1∗ − γ 2 B2 B2∗ + Y1 C1∗ C1 Y1 /γ 2 < 0. 274 H∞ CONTROL By Lemma 14.2, given X1 > 0 and Y1 > 0, there exists X12 and X2 such that Ỹ = γ 2 X̃ −1 or Ỹ /γ = (X̃/γ)−1 : #−1 " # " X1 /γ X12 /γ Y1 /γ ⋆ = ∗ X12 /γ X2 /γ ⋆ ⋆ if and only if " X1 /γ In In Y1 /γ # ≥0 rank " X1 /γ In In Y1 /γ # ≤ n + r. ✷ To show that the inequalities in the preceding lemma imply the existence of the stabilizing solutions to the Riccati equations of X∞ and Y∞ , we need the following theorem. Theorem 14.4 Let R ≥ 0 and suppose (A, R) is controllable and there is an X = X ∗ such that Q(X) := XA + A∗ X + XRX + Q < 0. (14.5) Then there exists a solution X+ > X to the Riccati equation X+ A + A∗ X+ + X+ RX+ + Q = 0 (14.6) such that A + RX+ is antistable. Proof. Let R = BB ∗ for some B. Note the fact that (A, R) is controllable iff (A, B) is. Let X be such that Q(X) < 0. Since (A, B) is controllable, there is an F0 such that A0 := A − BF0 is antistable. Now let X0 = X0∗ be the unique solution to the Lyapunov equation X0 A0 + A∗0 X0 − F0∗ F0 + Q = 0. Define F̂0 := F0 + B ∗ X, and we have the following equation: (X0 − X)A0 + A∗0 (X0 − X) = F̂0∗ F̂0 − Q(X) > 0. The antistability of A0 implies that X0 > X. Starting with X0 , we shall define a nonincreasing sequence of Hermitian matrices {Xi }. Associated with {Xi }, we shall also define a sequence of antistable matrices {Ai } and a 14.2. A Simplified H∞ Control Problem 275 sequence of matrices {Fi }. Assume inductively that we have already defined matrices {Xi }, {Ai }, and {Fi } for i up to n − 1 such that Xi is Hermitian and X0 ≥ X1 ≥ · · · ≥ Xn−1 > X, Ai = A − BFi is antistable, i = 0, . . . , n − 1; Fi = −B ∗ Xi−1 , i = 1, . . . , n − 1; Next, introduce Xi Ai + A∗i Xi = Fi∗ Fi − Q, i = 0, 1, . . . , n − 1. (14.7) Fn = −B ∗ Xn−1 , An = A − BFn . First we show that An is antistable. Using equation (14.7), with i = n − 1, we get Xn−1 An + A∗n Xn−1 + Q − Fn∗ Fn − (Fn − Fn−1 )∗ (Fn − Fn−1 ) = 0. (14.8) Let F̂n := Fn + B ∗ X; then (Xn−1 −X)An +A∗n (Xn−1 −X) = −Q(X)+F̂n∗ F̂n +(Fn −Fn−1 )∗ (Fn −Fn−1 ) > 0, (14.9) which implies that An is antistable by Lyapunov theorem since Xn−1 − X > 0. Now we introduce Xn as the unique solution of the Lyapunov equation: Xn An + A∗n Xn = Fn∗ Fn − Q. (14.10) Then Xn is Hermitian. Next, we have (Xn − X)An + A∗n (Xn − X) = −Q(X) + F̂n∗ F̂n > 0, and, by using equation (14.8), (Xn−1 − Xn )An + A∗n (Xn−1 − Xn ) = (Fn − Fn−1 )∗ (Fn − Fn−1 ) ≥ 0. Since An is antistable, we have Xn−1 ≥ Xn > X. We have a nonincreasing sequence {Xi }, and the sequence is bounded below by Xi > X. Hence the limit X+ := lim Xn n→∞ exists and is Hermitian, and we have X+ ≥ X. Passing the limit n → ∞ in equation (14.10), we get Q(X+ ) = 0. So X+ is a solution of equation (14.6). Note that X+ − X ≥ 0 and (X+ − X)A+ + A∗+ (X+ − X) = −Q(X) + (X+ − X)R(X+ − X) > 0. Hence, X+ − X > 0 and A+ = A + RX+ is antistable. (14.11) ✷ 276 H∞ CONTROL Lemma 14.5 There exists an admissible controller such that kTzw k∞ < γ only if the following three conditions hold: (i) There exists a stabilizing solution X∞ > 0 to X∞ A + A∗ X∞ + X∞ (B1 B1∗ /γ 2 − B2 B2∗ )X∞ + C1∗ C1 = 0. (14.12) (ii) There exists a stabilizing solution Y∞ > 0 to AY∞ + Y∞ A∗ + Y∞ (C1∗ C1 /γ 2 − C2∗ C2 )Y∞ + B1 B1∗ = 0. (iii) " −1 γY∞ In In −1 γX∞ # >0 or (14.13) ρ(X∞ Y∞ ) < γ 2 . Proof. Applying Theorem 14.4 to part (i) of Lemma 14.3, we conclude that there exists a Y > Y1 > 0 such that AY + Y A∗ + Y C1∗ C1 Y /γ 2 + B1 B1∗ − γ 2 B2 B2∗ = 0 and A + C1∗ C1 Y /γ 2 is antistable. Let X∞ := γ 2 Y −1 ; we have X∞ A + A∗ X∞ + X∞ (B1 B1∗ /γ 2 − B2 B2∗ )X∞ + C1∗ C1 = 0 (14.14) and −1 −1 −1 A + (B1 B1∗ /γ 2 − B2 B2∗ )X∞ = −X∞ (A + C1∗ C1 X∞ )X∞ = −X∞ (A + C1∗ C1 Y /γ 2 )X∞ is stable. Similarly, applying Theorem 14.4 to part (ii) of Lemma 14.3, we conclude that there exists an X > X1 > 0 such that XA + A∗ X + XB1 B1∗ X/γ 2 + C1∗ C1 − γ 2 C2∗ C2 = 0 and A + B1 B1∗ X/γ 2 is antistable. Let Y∞ := γ 2 X −1 , we have AY∞ + Y∞ A∗ + Y∞ (C1∗ C1 /γ 2 − C2∗ C2 )Y∞ + B1 B1∗ = 0 (14.15) and A + (C1∗ C1 /γ 2 − C2∗ C2 )Y∞ is stable. Finally, note that the rank condition in part (iii) of Lemma 14.3 is automatically satisfied by r ≥ n, and # # " # " " −1 X1 /γ In X/γ In γY∞ In ≥0 > = −1 In Y1 /γ In Y /γ In γX∞ or ρ(X∞ Y∞ ) < γ 2 . ✷ 14.2. A Simplified H∞ Control Problem 277 Proof of Theorem 14.1: To complete the proof, we only need to show that the controller Ksub given in Theorem 14.1 renders kTzw k∞ < γ. Note that the closed-loop transfer function with Ksub is given by   " # B1 A B2 F∞   B A c c −Z∞ L∞ D21  Â∞ Tzw =  .  =: C D  −Z∞ L∞ C2 c c C1 D12 F∞ 0 Define P = " −1 γ 2 Y∞ ∗ −1 −1 −γ 2 (Z∞ ) Y∞ −1 −1 −γ 2 Y∞ Z∞ −1 −1 γ 2 Y∞ Z∞ # . Then it is easy to show that P > 0 and P Ac + A∗c P + P Bc Bc∗ P/γ 2 + Cc∗ Cc = 0. Moreover, Ac + Bc Bc∗ P/γ 2 = " −1 A + B1 B1∗ Y∞ 0 −1 −1 B2 F∞ − B1 B1∗ Y∞ Z∞ A + B1 B1∗ X∞ /γ 2 + B2 F∞ # has no eigenvalues on the imaginary axis since A + B1 B1∗ X∞ /γ 2 + B2 F∞ is stable and −1 A + B1 B1∗ Y∞ is antistable. Thus, by Corollary 12.3, kTzw k∞ < γ. ✷ Remark 14.1 It is appropriate to point out that the conditions stated in Lemma 14.3 are, in fact, necessary and sufficient; see Gahinet and Apkarian [1994] and Gahinet [1996] for a linear matrix inequality (LMI) approach to the H∞ problem. But the necessity should be suitably interpreted. For example, if one finds an X1 > 0 and a Y1 > 0 satisfying conditions (i) and (ii) but not condition (iii), this does not imply that there is no admissible H∞ controller since there might be other X1 > 0 and Y1 > 0 that satisfy all three conditions. For example, consider γ = 1 and h i   1 −1 1 0  " # " #    0 1   G(s) =  0 .   1 0   h i 1 0 0 1 It is easy to check that X1 = Y1 = 0.5 satisfy (i) and (ii) but not (iii). Nevertheless, we shall show in the next section that γopt = 0.7321 and thus a suboptimal controller exists for γ = 1. In fact, we can check that 1 < X1 < 2, 1 < Y1 < 2 also satisfy (i), (ii) and (iii). ✸ 278 H∞ CONTROL Example 14.1 Consider the feedback system shown in Figure 6.3 with P = 50(s + 1.4) , (s + 1)(s + 2) We = We shall design a controller so that the H∞ minimized. Note that " # " e We (I + P K)−1 = ũ −Wu K(I + P K)−1 2 , s + 0.2 s+1 . s + 10 " # " # d e norm from w = to z = is di ũ Wu = We (I + P K)−1 P −Wu K(I + P K)−1 P #" d di Then the problem can be set up in an LFT framework with  −0.2 2 2 0  −1 0 0  0     0 0 −2 0 We We P −We P     G(s) =  0 0 0 −10 −Wu  =  0 0   I P −P 0 0 0  1  0 0 −3  0 0 1 1 0 # =: Tzw 2 0 0 20 0 30 0 0 0 0 1 0 0 0 " d di 0 −20 −30 −3 # .        .  0    −1  0 A suboptimal H∞ controller can be computed by using the following command: ≫ [K, Tzw , γsubopt ] = hinfsyn(G, ny , nu , γmin , γmax , tol) where ny and nu are the dimensions of y and u; γmin and γmax are, respectively a lower bound and an upper bound for γopt ; and tol is a tolerance to the optimal value. Set ny = 1, nu = 1, γmin = 0, γmax = 10, tol = 0.0001; we get γsubopt = 0.7849 and a suboptimal controller K= 12.82(s/10 + 1)(s/7.27 + 1)(s/1.4 + 1) . (s/32449447.67 + 1)(s/22.19 + 1)(s/1.4 + 1)(s/0.2 + 1) If we set tol = 0.01, we would get γsubopt = 0.7875 and a suboptimal controller K̃ = 12.78(s/10 + 1)(s/7.27 + 1)(s/1.4 + 1) . (s/2335.59 + 1)(s/21.97 + 1)(s/1.4 + 1)(s/0.2 + 1) The only significant difference between K and K̃ is the exact location of the far-away stable controller pole. Figure 14.1 shows the closed-loop frequency response of σ (Tzw ) and Figure 14.2 shows the frequency responses of S, T, KS, and SP . 14.2. A Simplified H∞ Control Problem 279 0 10 −1 10 −2 −1 10 10 0 10 1 10 frequency (rad/sec) 2 10 3 10 4 10 Figure 14.1: The closed-loop frequency responses of σ(Tzw ) with K (solid line) and K̃ (dashed line) 1 10 T 0 10 KS −1 SP 10 S −2 10 −3 10 −2 10 −1 10 0 10 1 10 frequency (rad/sec) 2 10 3 10 4 10 Figure 14.2: The frequency responses of S, T, KS, and SP with K 280 H∞ CONTROL Example 14.2 Consider again the two-mass/spring/damper system shown in Figure 4.2. Assume that F1 is the control force, F2 is the disturbance force, and the measurements of x1 and x2 are corrupted by measurement noise: y= " y1 y2 # = " x1 x2 # + Wn " n1 n2 # ,  0.01(s + 10)  s + 100 Wn =  0  0  0.01(s + 10)  . s + 100 Our objective is to design a control law so that the effect of the disturbance force F2 on the positions of the two masses, x1 and x2 , are reduced in a frequency " range 0 ≤#ω ≤ 2. W1 0 The problem can be set up as shown in Figure 14.3, where We = is the 0 W2 performance weight and Wu is the control weight. In order to limit the control force, we shall choose s+5 . Wu = s + 50 ✻z2 Wu y ✻ ✲ K ✲ u = F1 w1 = F2 ❄ " x1 x2 # Plant ✲ We ✲ z1 w2 = ❄ ✛ e Wn ✛ " n1 n2 # Figure 14.3: Rejecting the disturbance force F2 by a feedback control  F2   Now let u = F1 , w =  n1 ; then the problem can be formulated in an LFT form n2 with # " #   " We P2 We P1 0    0i Wu G(s) =    h 0 P2 P1 Wn  14.2. A Simplified H∞ Control Problem 281 where P1 and P2 denote the transfer matrices from F1 and F2 to Let W1 = " x1 x2 # , respectively. 5 , W2 = 0. s/2 + 1 That is, we only want to reject the effect of the disturbance force F2 on the position x1 . Then the optimal H2 performance is kFℓ (G, K2 )k2 = 2.6584 and the H∞ performance with the optimal H2 controller is kFℓ (G, K2 )k∞ = 2.6079 while the optimal H∞ performance with an H∞ controller is kFℓ (G, K∞ )k∞ = 1.6101. This means that the effect of the disturbance force F2 in the desired frequency rang 0 ≤ ω ≤ 2 will be effectively reduced with the H∞ controller K∞ by 5/1.6101 = 3.1054 times at x1 . On the other hand, let 5 . W1 = 0, W2 = s/2 + 1 That is, we only want to reject the effect of the disturbance force F2 on the position x2 . Then the optimal H2 performance is kFℓ (G, K2 )k2 = 0.1659 and the H∞ performance with the optimal H2 controller is kFℓ (G, K2 )k∞ = 0.5202 while the optimal H∞ performance with an H∞ controller is kFℓ (G, K∞ )k∞ = 0.5189. This means that the effect of the disturbance force F2 in the desired frequency rang 0 ≤ ω ≤ 2 will be effectively reduced with the H∞ controller K∞ by 5/0.5189 = 9.6358 times at x2 . 1 10 The largest singular value H∞ Control H2 Control 0 10 −1 10 −1 10 0 10 frequency (rad/sec) 1 10 Figure 14.4: The largest singular value plot of the closed-loop system Tzw with an H2 controller and an H∞ controller Finally, set W1 = W2 = 5 . s/2 + 1 282 H∞ CONTROL That is, we want to reject the effect of the disturbance force F2 on both x1 and x2 . Then the optimal H2 performance is kFℓ (G, K2 )k2 = 4.087 and the H∞ performance with the optimal H2 controller is kFℓ (G, K2 )k∞ = 6.0921 while the optimal H∞ performance with an H∞ controller is kFℓ (G, K∞ )k∞ = 4.3611. This means that the effect of the disturbance force F2 in the desired frequency rang 0 ≤ ω ≤ 2 will only be effectively reduced with the H∞ controller K∞ by 5/4.3611 = 1.1465 times at both x1 and x2 . This result shows clearly that it is very hard to reject the disturbance effect on both positions at the same time. The largest singular value Bode plots of the closed-loop system are shown in Figure 14.4. We note that the H∞ controller typically gives a relatively flat frequency response since it tries to minimize the peak of the frequency response. On the other hand, the H2 controller would typically produce a frequency response that rolls off fast in the high-frequency range but with a large peak in the low-frequency range. 14.3 Optimality and Limiting Behavior In this section, we will discuss, without proof, the behavior of the H∞ suboptimal solution as γ varies, especially as γ approaches the infima achievable norm, denoted by γopt . Since Theorem 14.1 gives necessary and sufficient conditions for the existence of an admissible controller such that kTzw k∞ < γ, γopt is the infimum over all γ such that conditions (i)–(iii) are satisfied. Theorem 14.1 does not give an explicit formula for γopt , but, just as for the H∞ norm calculation, it can be computed as closely as desired by a search technique. Although we have not focused on the problem of H∞ optimal controllers, the assumptions in this book make them relatively easy to obtain in most cases. In addition to describing the qualitative behavior of suboptimal solutions as γ varies, we will indicate why the descriptor version of the controller formulas below can usually provide formulas for the optimal controller when γ = γopt . As γ → ∞, H∞ → H2 , X∞ → X2 , etc., and Ksub → K2 . This fact is the result of the particular choice of the suboptimal controller. While it could be argued that Ksub is a natural choice, this connection with H2 actually hints at deeper interpretations. In fact, Ksub is the minimum entropy solution (see Section 14.4) as well as the minimax controller for kzk22 − γ 2 kwk22 . If γ2 ≥ γ1 > γopt , then X∞ (γ1 ) ≥ X∞ (γ2 ) and Y∞ (γ1 ) ≥ Y∞ (γ2 ). Thus X∞ and Y∞ are decreasing functions of γ, as is ρ(X∞ Y∞ ). At γ = γopt , any one of the three conditions in Theorem 14.1 can fail. If only condition (iii) fails, then it is relatively straightforward to show that the descriptor formulas below for γ = γopt are optimal; that is, the optimal controller is given by −2 (I − γopt Y∞ X∞ )x̂˙ u = As x̂ − L∞ y = F∞ x̂ (14.16) (14.17) 14.3. Optimality and Limiting Behavior 283 −2 −2 −2 where As := A + B2 F∞ + L∞ C2 + γopt Y∞ A∗ X∞ + γopt B1 B1∗ X∞ + γopt Y∞ C1∗ C1 . (See Example 14.3.) The formulas in Theorem 14.1 are not well-defined in the optimal case because the −2 term (I − γopt X∞ Y∞ ) is not invertible. It is possible but far less likely that conditions (i) or (ii) would fail before (iii). To see this, consider (i) and let γ1 be the largest γ for which H∞ fails to be in dom(Ric) because the H∞ matrix fails to have either the stability property or the complementarity property. The same remarks will apply to (ii) by duality. If complementarity fails at γ = γ1 , then ρ(X∞ ) → ∞ as γ → γ1 . For γ < γ1 , H∞ may again be in dom(Ric), but X∞ will be indefinite. For such γ, the controller u = −B2∗ X∞ x would make kTzw k∞ < γ but would not be stabilizing. (See part 1 of Example 14.3.) If the stability property fails at γ = γ1 , then H∞ 6∈ dom(Ric) but Ric can be extended to obtain X∞ so that a controller can be obtained to make kTzw k∞ = γ1 . The stability property will also not hold for any γ ≤ γ1 , and no controller whatsoever exists that makes kTzw k∞ < γ1 . In other words, if stability breaks down first, then the infimum over stabilizing controllers equals the infimum over all controllers, stabilizing or otherwise. (See part 2 of Example 14.3.) In view of this, we would typically expect that complementarity would fail first. Complementarity failing at γ = γ1 means ρ(X∞ ) → ∞ as γ → γ1 , so condition (iii) would fail at even larger values of γ, unless the eigenvectors associated with ρ(X∞ ) as γ → γ1 are in the null space of Y∞ . Thus condition (iii) is the most likely of all to fail first. If condition (i) or (ii) fails first because the stability property fails, the formulas in Theorem 14.1 as well as their descriptor versions are optimal at γ = γopt . This is illustrated in Example 14.3. If the complementarity condition fails first, [but (iii) does not fail], then obtaining formulas for the optimal controllers is a more subtle problem. Example 14.3 Let an interconnected dynamical system realization be given by h i   1 a 1 0  " # " #    0 1   G(s) =  0 .   1 0   h i 1 0 0 1 Then all assumptions for output feedback problem are satisfied and # " # " 2 1−γ 2 a a 1−γ 2 2 γ γ , J∞ = . H∞ = −1 −a −1 −a The eigenvalues of H∞ and J∞ are given by ) ( p (a2 + 1)γ 2 − 1 . ± γ 284 H∞ CONTROL 1 , then X− (H∞ ) and X− (J∞ ) exist and If γ > √ 2 a +1 " √ 2 2 (a +1)γ −1−aγ γ X− (H∞ ) = Im X− (J∞ ) = Im 1 " √ (a2 +1)γ 2 −1−aγ γ 1 # # . We shall consider two cases: 1) a > 0: In this case, the complementary property of dom(Ric) will fail before the stability property fails since p (a2 + 1)γ 2 − 1 − aγ = 0 when γ = 1. 1 and γ 6= 1, then H∞ ∈ dom(Ric) and Nevertheless, if γ > √ 2 a +1 ( > 0; if γ > 1 γ X∞ = p = 2 2 < 0; if √a12 +1 < γ < 1. (a + 1)γ − 1 − aγ Note that if γ > 1, then H∞ ∈ dom(Ric), J∞ ∈ dom(Ric), and γ X∞ = p >0 (a2 + 1)γ 2 − 1 − aγ γ > 0. Y∞ = p 2 (a + 1)γ 2 − 1 − aγ Hence conditions (i) and (ii) in Theorem 14.1 are satisfied, and we need to check condition (iii). Since γ2 ρ(X∞ Y∞ ) = p , ( (a2 + 1)γ 2 − 1 − aγ)2 it is clear that ρ(X∞ Y∞ ) → ∞ when γ → 1. So condition (iii) will fail before condition (i) or (ii) fails. 2) a < 0: In this case, the complementary property is always satisfied, and, furthermore, H∞ ∈ dom(Ric), J∞ ∈ dom(Ric), and γ X∞ = p >0 (a2 + 1)γ 2 − 1 − aγ 14.3. Optimality and Limiting Behavior 285 γ >0 Y∞ = p (a2 + 1)γ 2 − 1 − aγ 1 for γ > √ . 2 a +1 1 However, for γ ≤ √ , H∞ 6∈ dom(Ric) since stability property fails. Neva2 + 1 1 ertheless, in this case, if γ0 = √ , we can extend the dom(Ric) to include a2 + 1 those matrices H∞ with imaginary axis eigenvalues as " # −a X− (H∞ ) = Im 1 such that X∞ = − 1 is a solution to the Riccati equation a A∗ X∞ + X∞ A + C1∗ C1 + γ0−2 X∞ B1 B1∗ X∞ − X∞ B2 B2∗ X∞ = 0 and A + γ0−2 B1 B1∗ X∞ − B2 B2∗ X∞ = 0. It can be shown that γ2 ρ(X∞ Y∞ ) = p < γ2 ( (a2 + 1)γ 2 − 1 − aγ)2 is satisfied if and only if γ>   p 1 a2 + 2 + a > √ . a2 + 1 So condition (iii) of Theorem 14.1 will fail before either (i) or (ii) fails. In both a > 0 and a < 0 cases, the optimal γ for the output feedback is given by p γopt = a2 + 2 + a and the optimal controller given by the descriptor formula in equations (14.16) and (14.17) is a constant. In fact, γopt uopt = − q y. 2 − 1 − aγ 2 (a + 1)γopt opt √ For instance, let a = −1 then γopt = 3 − 1 = 0.7321 and uopt = −0.7321 y. Further,   −1.7321 1 −0.7321   Tzw =  1 0 0 . −0.7321 0 It is easy to check that kTzw k∞ = 0.7321. −0.7321 286 14.4 H∞ CONTROL Minimum Entropy Controller Let T be a transfer matrix with kT k∞ < γ. Then the entropy of T (s) is defined by γ2 I(T, γ) = − 2π Z ∞ γ2 2π Z −∞ It is easy to see that I(T, γ) = −
ln det I − γ −2 T ∗ (jω)T (jω) dω. ∞ −∞ X i ln 1 − γ −2 σi2 (T (jω)) dω and I(T, γ) ≥ 0, where σi (T (jω)) is the ith singular value of T (jω). It is also easy to show that Z ∞X 1 2 σ2 (T (jω)) dω = kT k2 . lim I(T, γ) = γ→∞ 2π −∞ i i Thus the entropy I(T, γ) is, in fact , a performance index measuring the tradeoff between the H∞ optimality (γ → kT k∞ ) and the H2 optimality (γ → ∞). It has been shown in Glover and Mustafa [1989] that the suboptimal controller given in Theorem 14.1 is actually the controller that satisfies the norm condition kTzw k∞ < γ and minimizes the following entropy: Z
γ2 ∞ ∗ ln det I − γ −2 Tzw (jω)Tzw (jω) dω. − 2π −∞ Therefore, the given suboptimal controller is also called the minimum entropy controller ˜ γ) = −I(T, γ)]. [maximum entropy controller if the entropy is defined as I(T, Related MATLAB Commands: hinfsyne, hinffi 14.5 An Optimal Controller To offer a general idea about the appearance of an optimal controller, we shall give in the following (without proof) the conditions under which an optimal controller exists and an explicit formula for an optimal controller. Theorem 14.6 There exists an admissible controller such that kTzw k∞ ≤ γ iff the following three conditions hold: (i) There exists a full column rank matrix # " X∞1 ∈ R2n×n X∞2 14.5. An Optimal Controller such that H∞ " X∞1 X∞2 287 # = " # X∞1 X∞2 TX , Re λi (TX ) ≤ 0 ∀i and ∗ ∗ X∞1 X∞2 = X∞2 X∞1 ; (ii) There exists a full column rank matrix " such that J∞ " Y∞1 Y∞2 # = " Y∞1 Y∞2 Y∞1 Y∞2 # ∈ R2n×n # TY , Re λi (TY ) ≤ 0 ∀i and ∗ ∗ Y∞1 Y∞2 = Y∞2 Y∞1 ; (iii) " ∗ X∞1 X∞2 ∗ X∞2 γ −1 Y∞2 ∗ Y∞2 γ −1 X∞2 ∗ Y∞1 Y∞2 # ≥ 0. Moreover, when these conditions hold, one such controller is Kopt (s) := CK (sEK − AK )+ BK where EK BK CK AK ∗ ∗ := Y∞1 X∞1 − γ −2 Y∞2 X∞2 ∗ ∗ := Y∞2 C2 := −B2∗ X∞2 ∗ := EK TX − BK C2 X∞1 = TY∗ EK + Y∞1 B2 CK . Remark 14.2 It is simple to show that if X∞1 and Y∞1 are nonsingular and if −1 −1 X∞ = X∞2 X∞1 and Y∞ = Y∞2 Y∞1 , then condition (iii) in the preceding theorem is equivalent to X∞ ≥ 0, Y∞ ≥ 0, and ρ(Y∞ X∞ ) ≤ γ 2 . So, in this case, the conditions for the existence of an optimal controller can be obtained from “taking the limit” of the corresponding conditions in Theorem 14.1. Moreover, the controller given above is reduced to the descriptor form given in equations (14.16) and (14.17). ✸ 288 14.6 H∞ CONTROL General H∞ Solutions Consider the system described by the block diagram ✛z ✛w G ✛ y ✲ K u where, as usual, G and K are assumed to be real rational and proper with K constrained to provide internal stability. The controller is said to be admissible if it is real rational, proper, and stabilizing. Although we are taking everything to be real, the results presented here are still true for the complex case with some obvious modifications. We will again only be interested in characterizing all suboptimal H∞ controllers. The realization of the transfer matrix G is taken to be of the form   " # A B1 B2 A B   G(s) =  C1 D11 D12  = , C D 0 C2 D21 which is compatible with the dimensions of z(t) ∈ Rp1 , y(t) ∈ Rp2 , w(t) ∈ Rm1 , u(t) ∈ Rm2 , and the state x(t) ∈ Rn . The following assumptions are made: (A1) (A, B2 ) is stabilizable and (C2 , A) is detectable; " # h i 0 (A2) D12 = and D21 = 0 I ; I " # A − jωI B2 (A3) has full column rank for all ω; C1 D12 " # A − jωI B1 (A4) has full row rank for all ω. C2 D21 Assumption (A1) is necessary for the existence of stabilizing controllers. The assumptions in (A2) mean that the penalty on z = C1 x + D12 u includes a nonsingular, normalized penalty on the control u, that the exogenous signal w includes both plant disturbance and sensor noise, and that the sensor noise weighting is normalized and nonsingular. Relaxation of (A2) leads to singular control problems; see Stroorvogel [1992]. For those problems that have D12 full column rank and D21 full row rank but do not satisfy assumption (A2), a normalizing procedure is given in the next section so that an equivalent new system will satisfy this assumption. 14.6. General H∞ Solutions 289 Assumptions (A3) and (A4) are made for a technical reason: Together with (A1) they guarantee that the two Hamiltonian matrices in the corresponding H2 problem belong to dom(Ric), as we have seen in Chapter 13. Dropping (A3) and (A4) would make the solution very complicated. A further discussion of the assumptions and their possible relaxation will be provided in Section 14.7. The main result is now stated in terms of the solutions of the X∞ and Y∞ Riccati equations together with the “state feedback” and “output injection” matrices F and L. R R̃ H∞ J∞ ∗ := D1• D1• − " γ 2 Im1 0 0 0 # , where D1• := [D11 D12 ] # " # D11 γ 2 Ip1 0 , where D•1 := := − 0 0 D21 " # " # i h A 0 B −1 ∗ ∗ := − R D C B 1 1• −C1∗ C1 −A∗ −C1∗ D1• " # " # h i A∗ 0 C∗ −1 ∗ := − R̃ D B C •1 1 ∗ −B1 B1∗ −A −B1 D•1 ∗ D•1 D•1 " X∞ := Ric(H∞ ) := " L := h F F1∞ F2∞ L1∞ # Y∞ := Ric(J∞ ) ∗ := −R−1 [D1• C1 + B ∗ X∞ ] L2∞ i ∗ := −[B1 D•1 + Y∞ C ∗ ]R̃−1 Partition D, F1∞ , and L1∞ are as follows:  ∗ F11∞ #  "  L∗11∞ D1111 F′  =  L∗ L′ D  12∞ D1121 0 L∗2∞ ∗ F12∞ ∗ F2∞ D1112 D1122 I 0 I 0    .   Remark 14.3 In the above matrix partitioning, some matrices may not exist depending on whether D12 or D21 is square. This issue will be discussed further later. For the time being, we shall assume that all matrices in the partition exist. ✸ Theorem 14.7 Suppose G satisfies the assumptions (A1)–(A4). (a) There exists an admissible controller K(s) such that ||Fℓ (G, K)||∞ < γ (i.e., kTzw k∞ < γ) if and only if 290 H∞ CONTROL ∗ ∗ (i) γ > max(σ[D1111 , D1112 , ], σ[D1111 , D1121 ]); (ii) H∞ ∈ dom(Ric) with X∞ = Ric(H∞ ) ≥ 0; (iii) J∞ ∈ dom(Ric) with Y∞ = Ric(J∞ ) ≥ 0; (iv) ρ(X∞ Y∞ ) < γ 2 . (b) Given that the conditions of part (a) are satisfied, then all rational internally stabilizing controllers K(s) satisfying ||Fℓ (G, K)||∞ < γ are given by K = Fℓ (M∞ , Q) for arbitrary Q ∈ RH∞ where  such that kQk∞ < γ   B̂1 B̂2  M∞ =   Ĉ1 Ĉ2 D̂11 D̂21  D̂12   0 ∗ ∗ D̂11 = −D1121 D1111 (γ 2 I − D1111 D1111 )−1 D1112 − D1122 D̂12 ∈ Rm2 ×m2 and D̂21 ∈ Rp2 ×p2 are any matrices (e.g., Cholesky factors) satisfying ∗ ∗ = I − D1121 (γ 2 I − D1111 D1111 )−1 D1121 , ∗ 2 ∗ −1 = I − D1112 (γ I − D1111 D1111 ) D1112 , ∗ D̂12 D̂12 ∗ D̂21 D̂21 and B̂2 Ĉ2 B̂1 Ĉ1 = Z∞ (B2 + L12∞ )D̂12 , = −D̂21 (C2 + F12∞ ), −1 D̂11 , = −Z∞ L2∞ + B̂2 D̂12 −1 Ĉ2 , = F2∞ + D̂11 D̂21 −1 Ĉ2  = A + BF + B̂1 D̂21 where Z∞ = (I − γ −2 Y∞ X∞ )−1 . (Note that if D11 = 0 then the formulas are considerably simplified.) Some Special Cases: Case 1: D12 = I In this case 1. In part (a), (i) becomes γ > σ(D1121 ). 14.7. Relaxing Assumptions 291 2. In part (b) D̂11 ∗ D̂12 D̂12 ∗ D̂21 D̂21 = −D1122 ∗ = I − γ −2 D1121 D1121 = I. Case 2: D21 = I In this case 1. In part (a), (i) becomes γ > σ(D1112 ). 2. In part (b) D̂11 ∗ D̂12 D̂12 = −D1122 = I ∗ D̂21 D̂21 ∗ D1112 . = I − γ −2 D1112 Case 3: D12 = I & D21 = I In this case 1. In part (a), (i) drops out. 2. In part (b) D̂11 ∗ D̂12 D̂12 ∗ D̂21 D̂21 14.7 = −D1122 = I = I. Relaxing Assumptions In this section, we indicate how the results of Section 14.6 can be extended to more general cases. Let a given problem have the following diagram, where zp (t) ∈ Rp1 , yp (t) ∈ Rp2 , wp (t) ∈ Rm1 , and up (t) ∈ Rm2 : ✛wp ✛zp P yp ✲ Kp ✛ up 292 H∞ CONTROL The plant P has the following state-space realization with Dp12 full column rank and Dp21 full row rank:   Bp2 Ap Bp1   P (s) =  Cp1 Dp11 Dp12  . Cp2 Dp21 Dp22 The objective is to find all rational proper controllers Kp (s) that stabilize P and ||Fℓ (P, Kp )||∞ < γ. To solve this problem, we first transform it to the standard one treated in the last section. Note that the following procedure can also be applied to the H2 problem (except the procedure for the case D11 6= 0). Normalize D12 and D21 Perform singular value decompositions to obtain the matrix factorizations # " h i 0 Rp , Dp21 = R̃p 0 I Ũp Dp12 = Up I such that Up and Ũp are square and unitary and Rp and R̃p are square and invertible. Now let zp = Up z, wp = Ũp∗ w, yp = R̃p y, up = Rp−1 u and K(s) = Rp Kp (s)R̃p G(s) = "  Up∗ 0 0 R̃p−1 # P (s) " Ũp∗ 0 0 Rp−1 Bp1 Ũp∗ Ap # Bp2 Rp−1     U ∗C Up∗ Dp12 Rp−1  Up∗ Dp11 Ũp∗   p p1 ∗ −1 −1 −1 −1 R̃p Cp2 R̃p Dp21 Ũp R̃p Dp22 Rp   # " B2 A B1 A B   . =:  C1 D11 D12  = C D C2 D21 D22 = Then D12 = and the new system is as follows: " 0 I # D21 = h 0 I i , 14.7. Relaxing Assumptions 293 ✛z ✛w G ✛ y u ✲ K Furthermore, ||Fℓ (P, Kp )||α = ||Up Fℓ (G, K)Ũp ||α = kFℓ (G, K)kα for α = 2 or ∞ since Up and Ũp are unitary. Moreover, Assumptions (A1), (A3), and (A4) remain unaffected. Remove the Assumption D22 = 0 Suppose K(s) is a controller for G with D22 set to zero. Then the controller for D22 6= 0 is K(I + D22 K)−1 . Hence there is no loss of generality in assuming D22 = 0. Relax (A3) and (A4) Suppose that  0 0  G= 0 0 1 1 1   1  0 which violates both (A3) and (A4) and corresponds to the robust stabilization of an integrator. If the controller u = −ǫx, where ǫ > 0 is used, then Tzw = −ǫs , with kTzw k∞ = ǫ. s+ǫ Hence the norm can be made arbitrarily small as ǫ → 0, but ǫ = 0 is not admissible since it is not stabilizing. This may be thought of as a case where the H∞ optimum is not achieved on the set of admissible controllers. Of course, for this system, H∞ optimal control is a silly problem, although the suboptimal case is not obviously so. Relax (A1) If assumption (A1) is violated, then it is obvious that no admissible controllers exist. Suppose (A1) is relaxed to allow unstabilizable and/or undetectable modes on the jω axis and internal stability is also relaxed to also allow closed-loop jω axis poles, but (A2)–(A4) is still satisfied. It can be easily shown that under these conditions the closed-loop H∞ norm cannot be made finite and, in particular, that the unstabilizable and/or undetectable modes on the jω axis must show up as poles in the closed-loop system. 294 H∞ CONTROL Violate (A1) and either or both (A3) and (A4) Sensible control problems can be posed that violate (A1) and either or both (A3) and (A4). For example, cases when A has modes at s = 0 that are unstabilizable through B2 and/or undetectable through C2 arise when an integrator is included in a weight on a disturbance input or an error term. In these cases, either (A3) or (A4) are also violated, or the closed-loop H∞ norm cannot be made finite. In many applications, such problems can be reformulated so that the integrator occurs inside the loop (essentially using the internal model principle) and is hence detectable and stabilizable. We will show this process in the next section. An alternative is to introduce ǫ perturbations so that (A1), (A3), and (A4) are satisfied. Roughly speaking, this would produce sensible answers for sensible problems, but the behavior as ǫ → 0 could be problematic. Relax (A2) In the cases that either D12 is not full column rank or that D21 is not full row rank, improper controllers can give a bounded H∞ norm for Tzw , although the controllers will not be admissible by our definition. Such singular filtering and control problems have been well-studied in H2 theory and many of the same techniques go over to the H∞ case (e.g., Willems [1981] and Willems, Kitapci, and Silverman [1986]). A complete solution to the singular problem can be found in Stroorvogel [1992]. 14.8 H2 and H∞ Integral Control It is interesting to note that the H2 and H∞ design frameworks do not, in general, produce integral control. In this section we show how to introduce integral control into the H2 and H∞ design framework through a simple disturbance rejection problem. We consider a feedback system shown in Figure 14.5. We shall assume that the frequency contents of the disturbance w are effectively modeled by the weighting Wd ∈ RH∞ and the constraints on control signal are limited by an appropriate choice of Wu ∈ RH∞ . In order to let the output y track the reference signal r, we require K to contain an integrator [i.e., K(s) has a pole at s = 0]. (In general, K is required to have poles on the imaginary axis.) There are several ways to achieve the integral design. One approach is to introduce an integral in the performance weight We . Then the transfer function between w and z1 is given by z1 = We (I + P K)−1 Wd w. Now if the resulting controller K stabilizes the plant P and makes the norm (2-norm or ∞-norm) between w and z1 finite, then K must have a pole at s = 0 that is the zero of the sensitivity function (assuming Wd has no zeros at s = 0). (This follows from the well-known internal model principle.) The problem with this approach is that the H∞ (or H2 ) control theory presented in this chapter and in the previous chapters cannot be 14.8. H2 and H∞ Integral Control 295 z2 ✻ w ❄ Wd Wu ✻ r ✲ g −✻ ✲ K u ❄ ✲ g ✲ P y ✲ We z ✲1 Figure 14.5: A simple disturbance rejection problem applied to this problem formulation directly because the pole s = 0 of We becomes an uncontrollable pole of the feedback system and the very first assumption for the H∞ (or H2 ) theory is violated. However, the obstacles can be overcome by appropriately reformulating the problem. Suppose We can be factorized as follows: We = W̃e (s)M (s) where M (s) is proper, containing all the imaginary axis poles of We , and M −1 (s) ∈ RH∞ , W̃e (s) is stable and minimum phase. Now suppose there exists a controller K(s) that contains the same imaginary axis poles that achieves the performance specifications. Then, without loss of generality, K can be factorized as K(s) = −K̂(s)M (s) such that there is no unstable pole/zero cancellation in forming the product K̂(s)M (s). Now the problem can be reformulated as in Figure 14.6. Figure 14.6 can be put in the general LFT framework as in Figure 14.7 with # " #   " W̃e M P W̃e M Wd   G(s) =  Wu 0 . M Wd MP We shall illustrate this design through a simple  0 1 s−2  = 3 2 P = (s + 1)(s − 3) −2 1 s + 10 Wu = = s + 100 " numerical example. Let  0  1  , Wd = 1, −100 −90 1 1 0 # , We = 1 . s 296 H∞ CONTROL ✻z2 w ❄ Wd Wu ✻ ✲ K̂ u ✲ ❄ g ✲ M ✲ P y1 ✲ W̃ e z ✲1 Figure 14.6: Disturbance rejection with imaginary axis poles z1 ✛ W̃e ✛ z2 ✛  Wu ✛  w Wd ✛  M ✛ ❄ g✛ P ✛ y1 u ✲ K̂ Figure 14.7: LFT framework for the disturbance rejection problem Then we can choose without loss of generality that M= s+α 1 , W̃e = , α > 0. s s+α This gives the following generalized system:  −α 0 1 −2 1   0 −100 0 0 0   0 0 0 −2α α    0 0 0 0 1 G(s) =   0 0 0 3 2    1 0 0 0 0   0 1 0 0 0  0 0 1 −2 1 1 0 α 0 0 0 −90 0 0 1 0 0 1 0 1 0         .        14.9. H∞ Filtering 297 The suboptimal H∞ controller K̂∞ for each α can be computed easily as K̂∞ = −2060381.4(s + 1)(s + α)(s + 100)(s − 0.1557) , (s + α)2 (s + 32.17)(s + 262343)(s − 19.89) which gives the closed-loop H∞ norm 7.854. Hence the controller K∞ = −K̂∞ (s)M (s) is given by K∞ (s) = 2060381.4(s + 1)(s + 100)(s − 0.1557) 7.85(s + 1)(s + 100)(s − 0.1557) ≈ , s(s + 32.17)(s + 262343)(s − 19.89) s(s + 32.17)(s − 19.89) which is independent of α as expected. Similarly, we can solve an optimal H2 controller K̂2 = −43.487(s + 1)(s + α)(s + 100)(s − 0.069) (s + α)2 (s2 + 30.94s + 411.81)(s − 7.964) and K2 (s) = −K̂2 (s)M (s) = 43.487(s + 1)(s + 100)(s − 0.069) . s(s2 + 30.94s + 411.81)(s − 7.964) An approximate integral control can also be achieved without going through the preceding process by letting We = W̃e = 1 , M (s) = 1 s+ǫ for a sufficiently small ǫ > 0. For example, a controller for ǫ = 0.001 is given by K∞ = 316880(s + 1)(s + 100)(s − 0.1545) 7.85(s + 1)(s + 100)(s − 0.1545) ≈ , (s + 0.001)(s + 32)(s + 40370)(s − 20) s(s + 32)(s − 20) which gives the closed-loop H∞ norm of 7.85. Similarly, an approximate H2 integral controller is obtained as K2 = 43.47(s + 1)(s + 100)(s − 0.0679) 43.47(s + 1)(s + 100)(s − 0.0679) ≈ . 2 (s + 0.001)(s + 30.93s + 411.7)(s − 7.972) s(s2 + 30.93s + 411.7)(s − 7.972) 14.9 H∞ Filtering In this section we show how the filtering problem can be solved using the H∞ theory developed earlier. Suppose a dynamic system is described by the following equations: ẋ = Ax + B1 w(t), x(0) = 0 y = C2 x + D21 w(t) z = C1 x + D11 w(t) (14.18) (14.19) (14.20) The filtering problem is to find an estimate ẑ of z in some sense using the measurement of y. The restriction on the filtering problem is that the filter has to be causal so 298 H∞ CONTROL that it can be realized (i.e., ẑ has to be generated by a causal system acting on the measurements). We will further restrict our filter to be unbiased; that is, given T > 0 the estimate ẑ(t) = 0 ∀t ∈ [0, T ] if y(t) = 0, ∀t ∈ [0, T ]. Now we can state our H∞ filtering problem. H∞ Filtering: Given a γ > 0, find a causal filter F (s) ∈ RH∞ if it exists such that 2 kz − ẑk2 J := sup < γ2 2 w∈L2 [0,∞) kwk2 with ẑ = F (s)y. A diagram for the filtering problem is shown in Figure 14.8. z z∆ ✛ ❄ ✛ j − ẑ F (s) ✛ y  A   C1 C2 B1 ✛ w  D11  D21 Figure 14.8: Filtering problem formulation The preceding filtering problem can also be formulated in an LFT framework: Given a system shown below ✛z∆ ✛w   0 A B1 G(s) ✛   G(s) =  C1 D11 −I  y ẑ C2 D21 0 ✲ F (s) find a filter F (s) ∈ RH∞ such that 2 sup w∈L2 kz∆ k2 2 kwk2 < γ2. (14.21) Hence the filtering problem can be regarded as a special H∞ problem. However, compared with control problems, there is no internal stability requirement in the filtering problem. Hence the solution to the above filtering problem can be obtained from the H∞ solution in the previous sections by setting B2 = 0 and dropping the internal stability requirement. Theorem 14.8 Suppose (C2 , A) is detectable and # " A − jωI B1 C2 D21 14.10. Notes and References 299 has full row rank for all ω. Let D21 be normalized and D11 partitioned conformably as " # " # D11 D111 D112 = . D21 0 I Then there exists a causal F (s) ∈ RH∞ such that J < γ 2 if and only if σ(D111 ) < γ and J∞ ∈ dom(Ric) with Y∞ = Ric(J∞ ) ≥ 0, where #∗ " #" # " D11 γ2I 0 D11 − R̃ := D21 D21 0 0 # " " # " # D11 B1∗ C1 A∗ 0 C1∗ C2∗ −1 . J∞ := − R̃ ∗ ∗ D21 B1∗ C2 −B1 B1∗ −A −B1 D11 −B1 D21 Moreover, if the above conditions are satisfied, then a rational causal filter F (s) satisfying J < γ 2 is given by " # A + L2∞ C2 + L1∞ D112 C2 −L2∞ − L1∞ D112 y ẑ = F (s)y = C1 − D112 C2 D112 where h L1∞ L2∞ i := − h ∗ B1 D11 + Y∞ C1∗ ∗ B1 D21 + Y∞ C2∗ i R̃−1 . ∗ In the case where D11 = 0 and B1 D21 = 0 the filter becomes much simpler: # " A − Y∞ C2∗ C2 Y∞ C2∗ y ẑ = C1 0 where Y∞ is the stabilizing solution to Y∞ A∗ + AY∞ + Y∞ (γ −2 C1∗ C1 − C2∗ C2 )Y∞ + B1 B1∗ = 0. 14.10 Notes and References The first part of this chapter is based on Sampei, Mita, and Nakamichi [1990], Packard [1994], Doyle, Glover, Khargonekar, and Francis [1989], and Chen and Zhou [1996]. The proof of Theorem 14.4 comes from Ran and Vreugdenhil [1988]. The detailed derivation of the H∞ solution for the general case is treated in Glover and Doyle [1988, 1989]. A fairly complete solution to the singular H∞ problem is obtained in Stoorvogel [1992]. The H∞ filtering and smoothing problems are considered in detail in Nagpal and Khargonekar [1991]. There is a rich body of literature on the LMI approach to H∞ control and related problems. In particular, readers are referred to the monograph by Boyd, Ghaoui, Feron, and Balakrishnan [1994] for a comprehensive treatment of LMIs and their applications in control. See also the paper by Chilali and Gahinet [1996] for an application of LMIs in H∞ control with closed-loop pole constraints. 300 H∞ CONTROL 14.11 Problems Problem 14.1 Figure 14.9 shows a single-loop analog feedback system. The plant is ✻ z1 ✻ z2 w2 ❄ ǫ2 W ǫ1 ✻ w1 e ✲ j −✻ ✻ ✲ F ❄y ✲ j ✲ K u ✲ P ✲ Figure 14.9: Analog feedback system P and the controller K; F is an antialiasing filter for future digital implementation of the controller (it is a good idea to include F at the start of the analog design so that there are no surprises later due to additional phase lag). The basic control specification is to get good tracking over a certain frequency range, say [0, ω1 ]; that is, to make the magnitude of the transfer function from w1 to e small over this frequency range. The weighted tracking error is z1 in the figure, where the weight W is selected to be a lowpass filter with bandwidth ω1 . We could attempt to minimize the H∞ norm from w1 to z1 , but this problem is not regular. To regularize it, another input, w2 , is added and another signal, z2 , is penalized. The two weights ǫ1 and ǫ2 are small positive scalars. The design problem is to minimize the H∞ norm " # " # w1 z1 from w = to z = . w2 z2 Figure 14.9 can then be converted to the standard setup by stacking the states of P , F , and W to form the state of G. The plant transfer function is taken to be P (s) = 20 − s . (s + 0.01)(s + 20) With a view toward subsequent digital control with h = 0.5, the filter F is taken to have bandwidth π/0.5, and Nyquist frequency ωN : F (s) = 1 . (0.5/π)s + 1 14.11. Problems 301 The weight W is then taken to have bandwidth one-fifth the Nyquist frequency: 2  1 . W (s) = (2.5/π)s + 1 Finally, ǫ1 and ǫ2 are both set to 0.01. Run hinfsyn and show your plots of the closed-loop frequency responses. Problem 14.2 Make the same assumptions as in Chapter 13 for H2 control and derive the H∞ controller parameterization by using the normalization procedure and Theorem 14.7. Problem 14.3 Consider the feedback system in Figure 6.3 and suppose P = s − 10 1 s+2 , We = , Wu = . (s + 1)(s + 10) s + 0.001 s + 10 Design a controller that minimizes " We So Wu KSo # . ∞ Simulate the time response of the system when r is a step. Problem 14.4 Repeat Problem 14.3 when We = 1/s. Problem 14.5 Consider again Problem 13.5 and design a controller that minimizes the H∞ norm of the transfer matrix from r to (e, uw ). Problem 14.6 Repeat Problem 14.5 with W = ǫ for ǫ = 0.01 and 0.0001. Study the behavior of the controller when ǫ → 0. Problem 14.7 Repeat Problem 14.5 and Problem 14.6 with P = 10(2 − s) . (s + 1)3 Problem 14.8 Let N ∈ RH− ∞ . The Nehari problem is the following approximation problem: inf kN − Qk∞ . Q∈RH∞ Formulate the Nehari problem as a standard H∞ control problem. Problem 14.9 Consider a generalized plant   −4 25 0.8 −1    −10 29 0.9 −1    P =  10 −25 0 1   1 0 13 25 302 H∞ CONTROL with an SISO controller K. Find the optimal H∞ performance γopt . Calculate the central controller for each γ ∈ (γopt , 2) and the corresponding H∞ performance, γ∞ . Plot γ∞ versus γ. Is γ∞ monotonic with respect to γ? (See Ushida and Kimura [1996] for a detailed discussion.) # " A B , where Problem 14.10 Let a satellite model be given by Po (s) = C 0     0 0 1 0 0     −5   0 0   0 0  , B =  1.7319 × 10 , A=  0 0    0 0 1     0 0 −1.5392 −2 × 0.003 × 1.539 3.7859 × 10−4 h i C = 1 0 1 0 , D = 0. Suppose the true system is described by P = (N + ∆N )(M + ∆M )−1 where Po = N M −1 is a normalized coprime factorization. Design a controller so that the controller stabilizes the largest " # ∆N ∆= . ∆M Problem 14.11 Consider the feedback system shown below and let P = 0.5(1 − s) , (s + 2)(s + 0.5) W1 = 50 ✲ W2 e ✲ K − ✻ s/1.245 + 1 , s/0.007 + 1 z1 ✲ ∆ W2 = 0.1256 s/0.502 + 1 . s/2 + 1 w2 w1 ❈ ❈ e ✲ W1 ✲❈❲ ❄ ✲ P z2 ✲ Figure 14.10: System with additive uncertainty Then " # z1 z2 = −1 where S = (I + P K) . " −W2 KS W1 S −W2 KS W1 S #" w1 w2 # =M " w1 w2 # 14.11. Problems 303 (a) Design a controller K such that inf K stabilizing kM k∞ . (b) Design a controller K so that inf sup µ∆ (M ), K stabilizing ω ∆= " ∆1 ∆2 # . Note that µ∆ (M ) = |W1 S| + |W2 KS|. Problem 14.12 Design a µ-synthesis controller for the HIMAT control problem in Example 9.1. Problem 14.13 Let G(s) ∈ H∞ be a square transfer matrix and α > 0. Show that G is (extended) strictly positive real (i.e., G∗ (jω) + G(jω) > 0, ∀ ω ∈ R ∪ {∞}) if and only if (αI − G)(αI + G)−1 ∞ < 1. Problem 14.14 Consider a generalized system  A B1  G(s) =  C1 D11 C2 D21 B2   D12  D22 and suppose we want to find a controller K such that Fℓ (G, K) is (extended) strictly postive real. Show that the problem can be converted to a standard H∞ control problem by using the transformation in the last problem. Problem 14.15 (Synthesis Using Popov Criterion) A stability criterion by Popov involves finding a controller and a multiplier matrix N such that Q + (I + sN )Fℓ (G, K) is strictly positive real where Q is a constant matrix. Assume D11 = 0 and D12 = 0 in the realization of G. Find the G̃ so that Q + (I + sN )Fℓ (G, K) = Fℓ (G̃, K) Hence the results in the last problem can be applied. 304 H∞ CONTROL Chapter 15 Controller Reduction We have shown in the previous chapters that the H∞ control theory and µ synthesis can be used to design robust performance controllers for highly complex uncertain systems. However, since a great many physical plants are modeled as high-order dynamical systems, the controllers designed with these methodologies typically have orders comparable to those of the plants. Simple linear controllers are normally preferred over complex linear controllers in control system designs for some obvious reasons: They are easier to understand and computationally less demanding; they are also easier to implement and have higher reliability since there are fewer things to go wrong in the hardware or bugs to fix in the software. Therefore, a lower-order controller should be sought whenever the resulting performance degradation is kept within an acceptable level. There are usually three ways to arrive at a lower-order controller. A seemingly obvious approach is to design lower-order controllers directly based on the high-order models. However, this is still largely an open research problem. Another approach is first to reduce the order of a high-order plant and, second, based on the reduced plant model, design a lower-order controller. A potential problem associated with this approach is that such a lower-order controller may not even stabilize the full-order plant since the error information between the full-order model and the reduced-order model is not considered in the design of the controller. On the other hand, one may seek to design first a high-order, high-performance controller and subsequently proceed with a reduction of the designed controller. This approach is usually referred to as controller reduction. A crucial consideration in controller order reduction is to take into account the closed loop so that closed-loop stability is guaranteed and the performance degradation is minimized with the reduced-order controllers. The purpose of this chapter is to introduce several controller reduction methods that can guarantee closed-loop stability and possibly closed-loop performance as well. 305 306 15.1 CONTROLLER REDUCTION H∞ Controller Reductions In this section, we consider an H∞ performance-preserving controller order reduction problem. We consider the feedback system shown in Figure 15.1 with a generalized plant realization given by   B2 A B1   G(s) =  C1 D11 D12  . C2 ✛ D21 z D22 ✛ G(s) w ✛ y u ✲ K(s) Figure 15.1: Closed-loop system diagram The following assumptions are made: (A1) (A, B2 ) is stabilizable and (C2 , A) is detectable; (A2) D12 has full " A − jωI (A3) C1 " A − jωI (A4) C2 column rank and D21 has full row rank; # B2 has full column rank for all ω; D12 # B1 has full row rank for all ω. D21 As stated in Chapter 14, all stabilizing controllers satisfying kTzw k∞ < γ can be parameterized as K = Fℓ (M∞ , Q), Q ∈ RH∞ , kQk∞ < γ (15.1) where M∞ is of the form M∞ = " M11 (s) M21 (s) M12 (s) M22 (s) #    B̂1 B̂2  =  Ĉ1 Ĉ2 D̂11 D̂21  D̂12  D̂22 −1 −1 Ĉ2 are both Ĉ1 and  − B̂1 D̂21 such that D̂12 and D̂21 are invertible and  − B̂2 D̂12 −1 −1 stable (i.e., M12 and M21 are both stable). 15.1. H∞ Controller Reductions 307 The problem to be considered here is to find a controller K̂ with a minimal possible order such that the H∞ performance requirement Fℓ (G, K̂) < γ is satisfied. This is ∞ clearly equivalent to finding a Q so that it satisfies the preceding constraint and the order of K̂ is minimized. Instead of choosing Q directly, we shall approach this problem from a different perspective. The following lemma is useful in the subsequent development and can be regarded as a special case of Theorem 10.6 (main loop theorem). Lemma 15.1 Consider a feedback system shown below ✛z ✛w N ✛ y ✲ Q u where N is a suitably partitioned transfer matrix # " N11 N12 . N (s) = N21 N22 Then the closed-loop transfer matrix from w to z is given by Tzw = Fℓ (N, Q) = N11 + N12 Q(I − N22 Q)−1 N21 . Assume that the feedback loop is well-posed [i.e., det(I − N22 (∞)Q(∞)) 6= 0] and either N21 (jω) has full row rank for all ω ∈ R ∪ ∞ or N12 (jω) has full column rank for all ω ∈ R ∪ ∞ and kN k∞ ≤ 1; then kFℓ (N, Q)k∞ < 1 if kQk∞ < 1. Proof. We shall assume that N21 has full row rank. The case when N12 has full column rank can be shown in the same way. To show that kTzw k∞ < 1, consider the closed-loop system at any frequency s = jω with the signals fixed as complex constant vectors. Let kQk∞ =: ǫ < 1 and note that + + Twy = N21 (I − N22 Q), where N21 is a right inverse of N21 . Also let κ := kTwy k∞ . Then kwk2 ≤ κkyk2, and kN k∞ ≤ 1 implies that kzk22 + kyk22 ≤ kwk22 + kuk22 . Therefore, kzk22 ≤ kwk22 + (ǫ2 − 1)kyk22 ≤ [1 − (1 − ǫ2 )κ−2 ]kwk22 , which implies kTzw k∞ < 1 . ✷ 308 15.1.1 CONTROLLER REDUCTION Additive Reduction Consider the class of (reduced-order) controllers that can be represented in the form K̂ = K0 + W2 ∆W1 , where K0 may be interpreted as a nominal, higher-order controller, and ∆ is a stable perturbation with stable, minimum phase and invertible weighting functions W1 and W2 . Suppose that kFℓ (G, K0 )k∞ < γ. A natural question is whether it is possible to obtain a reduced-order controller K̂ in this class such that the H∞ performance bound remains valid when K̂ is in place of K0 . Note that this is somewhat a special case of the preceding general problem: The specific form of K̂ means that K̂ and K0 must possess the same right-half plane poles, thus to a certain degree limiting the set of attainable reduced-order controllers. Suppose K̂ is a suboptimal H∞ controller; that is, there is a Q ∈ RH∞ with kQk∞ < γ such that K̂ = Fℓ (M∞ , Q). It follows from simple algebra that Q = Fℓ (K̄a−1 , K̂) where K̄a−1 := " 0 I I 0 # −1 M∞ " 0 I # I 0 . Furthermore, it follows from straightforward manipulations that kQk∞ < γ where R̃ = " γ −1/2 I 0 ⇐⇒ Fℓ (K̄a−1 , K̂) ⇐⇒ Fℓ (K̄a−1 , K0 + W2 ∆W1 ) ⇐⇒ Fℓ (R̃, ∆) 0 W1 #" R11 R21 ∞ R12 R22 ∞ <γ ∞ <γ <1 #" and R is given by the star product # " " Ko R11 R12 = S(K̄a−1 , R21 R22 I γ −1/2 I 0 I 0 # 0 W2 # ). It is easy to see that R̃12 and R̃21 are both minimum phase and invertible and hence have full column and full row rank, respectively, for all ω ∈ R ∪ ∞. Consequently, by invoking Lemma 15.1, we conclude that if R̃ is a contraction and k∆k∞ < 1, then Fℓ (R̃, ∆) ∞ < 1. This guarantees the existence of a Q such that kQk∞ < γ or, equivalently, the existence of a K̂ such that Fℓ (G, K̂) to the following theorem. ∞ < γ. This observation leads 15.1. H∞ Controller Reductions 309 Theorem 15.2 Suppose W1 and W2 are stable, minimum phase and invertible transfer matrices such that R̃ is a contraction. Let K0 be a stabilizing controller such that kFℓ (G, K0 )k∞ < γ. Then K̂ is also a stabilizing controller such that Fℓ (G, K̂) <γ ∞ if < 1. k∆k∞ = W2−1 (K̂ − K0 )W1−1 ∞ Since R̃ can always be made contractive for sufficiently small W1 and W2 , there are infinite many W1 and W2 that satisfy the conditions in the theorem. It is obvious that < 1 for some K̂, one would like to select the “largest” to make W2−1 (K̂ − K0 )W1−1 ∞ W1 and W2 such that R̃ is a contraction. Lemma 15.3 Assume that kR22 k∞ < γ and define  0 0 −R11 " #  ∼ ∼  −R11 L1 L2 R21 0  L= = F ( ℓ  R21 0 L3 L∼  0 2 ∼ ∼ R12 0 −R22 Then R̃ is a contraction if W1 and W2 satisfy " # " (W1∼ W1 )−1 0 L1 ≥ ∼ −1 L∼ 0 (W2 W2 ) 2  R12 0 −R22 0 L2 L3 #   −1  , γ I).   . Proof. See Goddard and Glover [1993]. ✷ An algorithm that maximizes det(W1∼ W1 ) det(W2 W2∼ ) has been developed by Goddard and Glover [1993]. The procedure below, devised directly from the preceding theorem, can be used to generate a required reduced-order controller that will preserve the closed-loop H∞ performance bound Fℓ (G, K̂) < γ. ∞ 1. Let K0 be a full-order controller such that kFℓ (G, K0 )k∞ < γ; 2. Compute W1 and W2 so that R̃ is a contraction; 3. Use the weighted model reduction method in Chapter 7 or any other methods to find a K̂ so that W2−1 (K̂ − K0 )W1−1 < 1. ∞ Note that all controller reduction methods introduced in this book are only sufficient; that is, there may be desired reduced-order controllers that cannot be found from the proposed procedures. 310 CONTROLLER REDUCTION 15.1.2 Coprime Factor Reduction The H∞ controller reduction problem can also be considered in the coprime factor framework. For that purpose, we need the following alternative representation of all admissible H∞ controllers: Lemma 15.4 The family of all admissible controllers such that kTzw k∞ < γ can also be written as K(s) = Fℓ (M∞ , Q) = (Θ11 Q + Θ12 )(Θ21 Q + Θ22 )−1 := U V −1 = (QΘ̃12 + Θ̃22 )−1 (QΘ̃11 + Θ̃21 ) := Ṽ −1 Ũ where Q ∈ RH∞ , kQk∞ < γ, and U V −1 and Ṽ −1 Ũ are, respectively, right and left coprime factorizations over RH∞ , and Θ= Θ̃ = " " Θ11 Θ21 Θ̃11 Θ̃21 Θ−1 Θ12 Θ22 # Θ̃12 Θ̃22   =   Θ̃−1 =  #   −1 =  Ĉ1 − D̂11 D̂21 Ĉ2 −1 −D̂21 Ĉ2  −1 Ĉ1  − B̂2 D̂12  −1 =  Ĉ2 − D̂22 D̂12 Ĉ1 −1 D̂12 Ĉ1 −1  − B̂2 D̂12 Ĉ1 −1 −D̂12 Ĉ1 −1 Ĉ1 Ĉ2 − D̂22 D̂12 −1 Ĉ2  − B̂1 D̂21 −1 D̂21 Ĉ2 −1 Ĉ2 Ĉ1 − D̂11 D̂21 −1 B̂1 D̂21 −1 D̂22 B̂2 − B̂1 D̂21 −1 Ĉ2  − B̂1 D̂21 −1 D̂22 D̂12 − D̂11 D̂21 −1 −D̂21 D̂22 −1  D̂11 D̂21  −1 D̂21 −1 D̂11 B̂1 − B̂2 D̂12 −1 −B̂2 D̂12 −1 D̂11 D̂21 − D̂22 D̂12 −1 D̂12 D̂11 −1 B̂2 D̂12 −1 D̂12 −1 D̂22 D̂12 −1 −B̂1 D̂21 −1 D̂21 −1 −D̂11 D̂21  −1  −D̂22 D̂12  −1 D̂12  −1 D̂11 B̂1 − B̂2 D̂12  −1 −D̂12 D̂11  −1 D̂21 − D̂22 D̂12 D̂11  −1 D̂22 B̂2 − B̂1 D̂21  −1 D̂21 D̂22 . −1 D̂12 − D̂11 D̂21 D̂22 Proof. The results follow immediately from Lemma 9.2. Theorem 15.5 Let K0 kFℓ (G, K0 )k∞ < γ and let " γ −1 I 0  ✷ = Θ12 Θ−1 be the central H∞ controller such that 22 Û , V̂ ∈ RH∞ with det V̂ (∞) 6= 0 be such that " # " #! # √ Θ12 Û 0 −1 − < 1/ 2. (15.2) Θ Θ22 V̂ I ∞ Then K̂ = Û V̂ −1 is also a stabilizing controller such that kFℓ (G, K̂)k∞ < γ. 15.1. H∞ Controller Reductions 311 Proof. Note that by Lemma 15.4, K is an admissible controller such that kTzw k∞ < γ if and only if there exists a Q ∈ RH∞ with kQk∞ < γ such that " # " # " # U Θ11 Q + Θ12 Q := =Θ (15.3) V Θ21 Q + Θ22 I and K = U V −1 . Hence, to show that K̂ = Û V̂ −1 with Û and V̂ satisfying equation (15.2) is also a stabilizing controller such that kFℓ (G, K̂)k∞ < γ, we need to show that there is another coprime factorization for K̂ = U V −1 and a Q ∈ RH∞ with kQk∞ < γ such that equation (15.3) is satisfied. Define " #! # " " # Θ12 Û γ −1 I 0 −1 − ∆ := Θ Θ22 V̂ 0 I and partition ∆ as ∆ := Then and " Û V̂ " # = " Θ12 Θ22 # −Θ Û(I − ∆V )−1 V̂ (I − ∆V )−1 # " " ∆U ∆V γI 0 =Θ " 0 I # . # ∆=Θ " −γ∆U I − ∆V −γ∆U (I − ∆V )−1 I # # . Define U := Û (I −∆V )−1 , V := V̂ (I −∆V )−1 , and Q := −γ∆U (I −∆V )−1 . Then U V −1 is another coprime factorization for K̂. To show that K̂ = U V −1 = Û V̂ −1 is a stabilizing controller such that kFℓ (G, K̂)k∞ < γ, we need to show that γ∆U (I − ∆V )−1 ∞ < γ or, equivalently, ∆U (I − ∆V )−1 ∞ < 1. Now ∆U (I − ∆V )−1  h i −1 ∆ I− 0 I ∆  h i  0 √ I 0 = Fℓ  √ h √ i  , 2∆ I/ 2 0 I/ 2 = h I 0  and, by Lemma 15.1, ∆U (I − ∆V )−1  0  √ I/ 2 ∞ h i < 1 since h 0 I i  0 √ i  I/ 2 312 CONTROLLER REDUCTION is a contraction and √ 2∆ ∞ < 1. ✷ Similarly, we have the following theorem: = Θ̃−1 22 Θ̃21 be the central H∞ controller such that ˆ ˆ kFℓ (G, K0 )k∞ < γ and let Ũ , Ṽ ∈ RH∞ with det Ṽˆ (∞) 6= 0 be such that " # i h i h −1 √ γ I 0 −1 ˆ Ṽˆ < 1/ 2. Θ̃ Θ̃21 Θ̃22 − Ũ 0 I ∞ Theorem 15.6 Let K0 −1 ˆ is also a stabilizing controller such that kF (G, K̂)k < γ. Then K̂ = Ṽˆ Ũ ℓ ∞ The preceding two theorems show that the sufficient conditions for H∞ controller reduction problem are equivalent to frequency-weighted H∞ model reduction problems. H∞ Controller Reduction Procedures −1 (i) Let K0 = Θ12 Θ−1 22 (= Θ̃22 Θ̃21 ) be a suboptimal H∞ central controller (Q = 0) such that kTzw k∞ < γ. −1 ˆ ) such that (ii) Find a reduced-order controller K̂ = Û V̂ −1 (or Ṽˆ Ũ #! # " " " # √ Û Θ γ −1 I 0 12 < 1/ 2 − Θ−1 Θ22 V̂ 0 I ∞ or h Θ̃21 Θ̃22 i − h ˆ Ũ Ṽˆ i Θ̃ −1 " γ −1 I 0 0 I # √ < 1/ 2. ∞ Then the closed-loop system with the reduced-order controller K̂ is stable and the performance is maintained with the reduced-order controller; that is, kTzw k∞ = Fℓ (G, K̂) 15.2 ∞ < γ. Notes and References The main results presented in this chapter are based on the work of Goddard and Glover [1993, 1994]. Other controller reduction methods include the stability-oriented controller reduction criterion proposed by Enns [1984a, 1984b]; the weighted and unweighted coprime factor controller reduction methods studied by Liu and Anderson [1986, 1990]; Liu, Anderson, and Ly [1990]; Anderson and Liu [1989]; and Anderson [1993]; the normalized H∞ controller reduction studied by Mustafa and Glover [1991]; the normalized coprime factor method studied by McFarlane and Glover [1990] in the H∞ loop-shaping setup; and the controller reduction in the ν-gap metric setup studied by Vinnicombe [1993b]. Lenz, Khargonekar, and Doyle [1987] have also proposed another H∞ controller reduction method with guaranteed performance for a class of H∞ problems. 15.3. Problems 15.3 313 Problems Problem 15.1 Find a lower-order controller for the system in Example 10.4 when γ = 2. Problem 15.2 Find a lower-order controller for Problem 14.3 when γ = 1.1γopt where γopt is the optimal norm. Problem 15.3 Find a lower-order controller for the HIMAT control problem in Problem 14.12 when γ = 1.1γopt where γopt is the optimal norm. Compare the controller reduction methods presented in this chapter with other available methods. Problem 15.4 Let G be a generalized plant and K be a stabilizing controller. Let ∆ = diag(∆p , ∆k )"be a #suitably dimensioned perturbation and let Tẑŵ be the transfer " # w z matrix from ŵ = to ẑ = in the following diagram: w1 z1 z ✛ G zw z1 W −1 ✛ ✲ K ✛ ✛ w ✲ f✛ w1 ✲ (K̂ − K)W Let W, W −1 ∈ H∞ be a given transfer matrix. Show that the following statements are equivalent: ! " # I 0 Tẑŵ < 1; 1. µ∆ 0 W −1 2. kFℓ (G, K)k∞ < 1 and W −1 Fu (Tẑŵ , ∆p ) ∞ < 1 for all σ(∆p ) ≤ 1; ! # " I 0 < 1 for all σ(∆k ) ≤ 1. Tẑŵ , ∆k 3. W −1 Tz1 w1 ∞ < 1 and Fℓ 0 W −1 ∞ Problem 15.5 In the part "3 of Problem# 15.4, if we ! let ∆k = (K̂ − K)W , then Tz1 w1 = I 0 Tẑŵ , ∆k = Fℓ (G, K̂). Thus K̂ stabilizes the G22 (I − KG22 )−1 and Fℓ 0 W −1 system and satisfies Fℓ (G, K̂) ∞ < 1 if k∆k k∞ = (K̂ − K)W ∞ ≤ 1 and part 2 of 314 CONTROLLER REDUCTION Problem 15.4 is satisfied by a controller K. Hence to reduce the order of the controller K, it is sufficient to solve a frequency-weighted model reduction problem if W can be calculated. In the single-input and single-output case, a “smallest” weighting function W (s) can be calculated using part 2 of Problem 15.4 as follows: |W (jω)| ≥ sup σ(∆p )≤1 |Fu (Tẑŵ (jω), ∆p )|. Repeat Problems 15.1 and 15.2 using the foregoing method. (Hint: W can be computed frequency by frequency using µ software and then fitted by a stable and minimum phase transfer function.) Problem 15.6 One way to generalize the method in Problem 15.5 to the MIMO case is to take a diagonal W W = diag(W1 , W2 , . . . , Wm ) and let Ŵi be computed from |Ŵi (jω)| ≥ sup σ(∆p )≤1 eTi Fu (Tẑŵ (jω), ∆p ) where ei is the ith unit vector. Next let α(s) be computed from |α(jω)| ≥ sup σ(∆p )≤1 Ŵ −1 Fu (Tẑŵ (jω), ∆p ) where Ŵ = diag(Ŵ1 , Ŵ2 , . . . , Ŵm ). Then a suitable W can be taken as W = αŴ . Apply this method to Problem 15.3. Problem 15.7 Generalize the procedures in Problems 15.5 and 15.6 to problems with additional structured uncertainty cases. (A more general case can be found in Yang and Packard [1995].) Chapter 16 H∞ Loop Shaping This chapter introduces a design technique that incorporates loop-shaping methods to obtain performance/robust stability tradeoffs, and a particular H∞ optimization problem to guarantee closed-loop stability and a level of robust stability at all frequencies. The proposed technique uses only the basic concept of loop-shaping methods, and then a robust stabilization controller for the normalized coprime factor perturbed system is used to construct the final controller. This chapter is arranged as follows: The H∞ theory is applied to solve the stabilization problem of a normalized coprime factor perturbed system in Section 16.1. The loop-shaping design procedure is described in Section 16.2. The theoretical justification for the loop-shaping design procedure is given in Section 16.3. Some further loop-shaping guidelines are given in Section 16.4. 16.1 Robust Stabilization of Coprime Factors In this section, we use the H∞ control theory developed in previous chapters to solve the robust stabilization of a left coprime factor perturbed plant given by ˜M, ∆ ˜N with M̃ , Ñ, ∆ ˜ M )−1 (Ñ + ∆ ˜ N) P∆ = (M̃ + ∆ i h ˜M ˜N ∆ ∈ RH∞ and < ǫ (see Figure 16.1). The transfer ∆ ∞ matrices (M̃ , Ñ ) are assumed to be a left coprime factorization of P (i.e., P = M̃ −1 Ñ ), and K internally stabilizes the nominal system. It has been shown in Chapter 8 that the system is robustly stable iff " # K ≤ 1/ǫ. (I + P K)−1 M̃ −1 I ∞ Finding a controller such that the above norm condition holds is an H∞ norm minimization problem that can be solved using H∞ theory developed in previous chapters. 315 316 H∞ LOOP SHAPING z1 r ✲ ∆ ˜N − ✲ f✛ ˜M ✛ ∆ z2 w ✲f ✲ K −✻ ✲ Ñ ✲❄ f y ✲ ✲ M̃ −1 Figure 16.1: Left coprime factor perturbed systems Suppose P has a stabilizable and detectable state-space realization given by " # A B P = C D and let L be a matrix such that A + LC is stable. Then a left coprime factorization of P = M̃ −1 Ñ is given by " # h i A + LC B + LD L Ñ M̃ = ZC ZD Z where Z can be any nonsingular matrix. In particular, we shall choose Z = (I + DD∗ )−1/2 if P = M̃ −1 Ñ is chosen to be a normalized left coprime factorization. Denote K̂ = −K. Then the system diagram can be put in an LFT form, as in Figure generalized plant  # " #   " A −LZ −1 # " # "  " 0 I  0 0   −1  G(s) =  M̃ P  =  C Z −1  −1 M̃ P C Z −1   B2 A B1   =:  C1 D11 D12  . C2 D21 D22 16.2, with the B I D D  #      To apply the H∞ control formulas in Chapter 14, we need to normalize the D12 and D21 first. Note that # " " # " # 1 1 D∗ (I + DD∗ )− 2 (I + D∗ D)− 2 I 0 1 ∗ 2 =U (I + D D) , where U = 1 1 −(I + DD∗ )− 2 D(I + D∗ D)− 2 D I 16.1. Robust Stabilization of Coprime Factors 317 w z1 ✛ z ✛ 2 M̃ −1 ✛ ❄ i✛ Ñ ✛ y u ✲ K̂ Figure 16.2: LFT diagram for coprime factor stabilization and U is a unitary matrix. Let 1 K̂ = (I + D∗ D)− 2 K̃Z " # " # z1 ẑ1 = U . z2 ẑ2 Then kTzw k∞ = kU ∗ Tzw k∞ = kTẑw k∞ and the problem becomes one of finding a controller K̃ so that kTẑw k∞ < γ with the following generalized plant: # " # " U∗ 0 I 0 Ĝ = G 1 0 Z 0 (I + D∗ D)− 2  A −LZ −1 # " #  " ∗ − 21 ∗ − 21 −1  C Z −(I + DD ) −(I + DD ) = 1 1  (I + D∗ D)− 2 D∗ C (I + D∗ D)− 2 D∗ Z −1  I ZC " B # 0 I 1 ZD(I + D∗ D)− 2    .   Now the formulas in Chapter 14 can be applied to Ĝ to obtain a controller K̃ and then 1 the K can be obtained from K = −(I + D∗ D)− 2 K̃Z. We shall leave the detail to the reader. In the sequel, we shall consider the case D = 0 and Z = I. In this case, we have γ > 1 and X∞ (A − LC ∗ LL∗ γ2C ∗C LC ) + (A − 2 ) X∞ − X∞ (BB ∗ − 2 )X∞ + 2 =0 −1 γ −1 γ −1 γ −1 γ2 Y∞ (A + LC)∗ + (A + LC)Y∞ − Y∞ C ∗ CY∞ = 0. 318 H∞ LOOP SHAPING It is clear that Y∞ = 0 is the stabilizing solution. Hence by the formulas in Chapter 14 we have i h i h L1∞ L2∞ = 0 L and Z∞ = I, D̂11 = 0, D̂12 = I, D̂21 The results are summarized in the following theorem. p γ2 − 1 I. = γ Theorem 16.1 Let D = 0 and let L be such that A + LC is stable. Then there exists a controller K such that " # K <γ (I + P K)−1 M̃ −1 I ∞ iff γ > 1 and there exists a stabilizing solution X∞ ≥ 0 solving X∞ (A − LC LC ∗ LL∗ γ2C ∗C ) + (A − 2 ) X∞ − X∞ (BB ∗ − 2 )X∞ + 2 = 0. −1 γ −1 γ −1 γ −1 γ2 Moreover, if the above conditions hold a central controller is given by # " A − BB ∗ X∞ + LC L K= . − B ∗ X∞ 0 It is clear that the existence of a robust stabilizing controller depends on the choice of the stabilizing matrix L (i.e., the choice of the coprime factorization). Now let Y ≥ 0 be the stabilizing solution to AY + Y A∗ − Y C ∗ CY + BB ∗ = 0 and let L = −Y C ∗ . Then the left coprime factorization (M̃ , Ñ ) given by " # h i A − Y C ∗ C B −Y C ∗ Ñ M̃ = C 0 I is a normalized left coprime factorization (see Chapter 12). Let k·kH denote the Hankel norm (i.e., the largest Hankel singular value). Then we have the following result. Corollary 16.2 Let D = 0 and L = −Y C ∗ , where Y ≥ 0 is the stabilizing solution to AY + Y A∗ − Y C ∗ CY + BB ∗ = 0. Then P = M̃ −1 Ñ is a normalized left coprime factorization and " # K 1 = p (I + P K)−1 M̃ −1 inf K stabilizing I 1 − λmax (Y Q) ∞  h i = 1− Ñ M̃ 2 H −1/2 =: γmin 16.1. Robust Stabilization of Coprime Factors 319 where Q is the solution to the following Lyapunov equation: Q(A − Y C ∗ C) + (A − Y C ∗ C)∗ Q + C ∗ C = 0. Moreover, if the preceding conditions hold, then for any γ > γmin a controller achieving " # K <γ (I + P K)−1 M̃ −1 I ∞ is given by K(s) = " where X∞ = A − BB ∗ X∞ − Y C ∗ C − B ∗ X∞ # −Y C ∗ 0  −1 γ2 γ2 Q I − Y Q . γ2 − 1 γ2 − 1 Proof. Note that the Hamiltonian matrix associated with X∞ is given by # " A + γ 21−1 Y C ∗ C −BB ∗ + γ 21−1 Y C ∗ CY . H∞ = 2 −(A + γ 21−1 Y C ∗ C)∗ − γ 2γ−1 C ∗ C Straightforward calculation shows that " # " 2 I I − γ 2γ−1 Y Hq H∞ = γ2 0 0 γ 2 −1 I where Hq = " A − Y C ∗C −C ∗ C #−1 2 − γ 2γ−1 Y γ2 γ 2 −1 I 0 −(A − Y C ∗ C)∗ # . It is clear that the stable invariant subspace of Hq is given by " # I X− (Hq ) = Im Q and the stable invariant subspace of H∞ is given by # " # " 2 2 I − γ 2γ−1 Y Q I − γ 2γ−1 Y X− (Hq ) = Im . X− (H∞ ) = γ2 γ2 0 γ 2 −1 I γ 2 −1 Q Hence there is a nonnegative definite stabilizing solution to the algebraic Riccati equation of X∞ if and only if γ2 YQ>0 I− 2 γ −1 320 H∞ LOOP SHAPING or 1 γ>p 1 − λmax (Y Q) and the solution, if it exists, is given by  −1 γ2 γ2 Q I − Y Q . γ2 − 1 γ2 − 1 X∞ = Note that iY and Q are the controllability Gramian and the observability Gramian ofi h h respectively. Therefore, we also have that the Hankel norm of Ñ M̃ Ñ M̃ p is λmax (Y Q). ✷ Corollary 16.3 Let P = M̃ −1 Ñ be a normalized left coprime factorization and ˜ M )−1 (Ñ + ∆ ˜ N) P∆ = (M̃ + ∆ with h ˜M ∆ ˜N ∆ i ∞ < ǫ. Then there is a robustly stabilizing controller for P∆ if and only if r h i 2 p ǫ ≤ 1 − λmax (Y Q) = 1 − . Ñ M̃ H The solutions to the normalized left coprime factorization stabilization problem are also solutions to a related H∞ problem, which is shown in the following lemma. Lemma 16.4 Let P = M̃ −1 Ñ be a normalized left coprime factorization. Then " # " # h i K K . = (I + P K)−1 I P (I + P K)−1 M̃ −1 I I ∞ ∞ Proof. Since (M̃ , Ñ) is a normalized left coprime factorization of P , we have h ih i∼ =I M̃ Ñ M̃ Ñ and h M̃ Ñ Using these equations, we have " i K I ∞ # = h M̃ Ñ i∼ ∞ (I + P K)−1 M̃ −1 ∞ = 1. 16.1. Robust Stabilization of Coprime Factors This implies " = " ≤ " = " ≤ " = " K I # K I # K I # (I + P K)−1 M̃ −1 K I # (I + P K)−1 K I # (I + P K)−1 M̃ −1 K I # (I + P K)−1 M̃ −1 (I + P K)−1 M̃ −1 h I h h M̃ M̃ P i Ñ ih Ñ i M̃ Ñ h ∞ i∼ M̃ Ñ ∞ i∼ ∞ ∞ h ∞ 321 M̃ Ñ i ∞ . ∞ −1 (I + P K) M̃ −1 " = ∞ K I # (I + P K)−1 h I P i . ∞ ✷ Combining Corollary 16.3 and Lemma 16.4, we have the following result. Corollary 16.5 A controller solves the normalized left coprime factor robust stabilization problem if and only if it solves the following H∞ control problem: " # h i I <γ (I + P K)−1 I P K ∞ and inf K stabilizing " I K # (I + P K)−1 h I P i 1 p 1 − λmax (Y Q)  h i = 1− Ñ M̃ = ∞ 2 H −1/2 . The solution Q can also be obtained in other ways. Let X ≥ 0 be the stabilizing solution to XA + A∗ X − XBB ∗ X + C ∗ C = 0. Then it is easy to verify that Q = (I + XY )−1 X. 322 H∞ LOOP SHAPING Hence  h 1 = 1− γmin = p Ñ 1 − λmax (Y Q) M̃ i 2 H −1/2 = p 1 + λmax (XY ). Similar results can be obtained if one starts with a normalized right coprime factorization. In fact, a rather strong relation between the normalized left and right coprime factorization problems can be established using the following matrix fact. Lemma 16.6 Let M be a square matrix such that M 2 = M . Then σi (M ) = σi (I − M ) for all i such that 0 < σi (M ) 6= 1. Proof. We first show that the eigenvalues of M are either 0 or 1 and M is diagonalizable. In fact, assume that λ is an eigenvalue of M and x is a corresponding eigenvector. Then λx = M x = M M x = M (M x) = λM x = λ2 x; that is, λ(1 − λ)x = 0. This implies that either λ = 0 or λ = 1. To show that M is diagonalizable, assume that M = T JT −1, where J is a Jordan canonical form. It follows immediately that J must be diagonal by the condition M = M 2 . Next, assume that M is diagonalized by a nonsingular matrix T such that " # I 0 M =T T −1 . 0 0 Then N := I − M = T Define " A B∗ B D " # 0 0 0 I # T −1 . := T ∗ T and assume 0 < λ 6= 1. Then A > 0 and ⇔ ⇔ det(M ∗ M − λI) = 0 " # " # I 0 I 0 det( T ∗T − λT ∗ T ) = 0 0 0 0 0 " # (1 − λ)A −λB det =0 −λB ∗ −λD ⇔ det(−λD − ⇔ det( λ2 B ∗ A−1 B) = 0 1−λ 1−λ D + B ∗ A−1 B) = 0 λ 16.1. Robust Stabilization of Coprime Factors " −λA −λB ∗ −λB (1 − λ)D ⇔ det ⇔ det(N ∗ N − λI) = 0. # 323 =0 This implies that all nonzero eigenvalues of M ∗ M and N ∗ N that are not equal to 1 are equal; that is, σi (M ) = σi (I − M ) for all i such that 0 < σi (M ) 6= 1. ✷ Using this matrix fact, we have the following corollary. Corollary 16.7 Let K and P be any compatibly dimensioned complex matrices. Then " # " # h i h i I I . (I + KP )−1 I K = (I + P K)−1 I P P K Proof. Define " # h I M= (I + P K)−1 I K P i , N= " −P I # (I + KP )−1 h −K I i . Then it is easy to verify that M 2 = M and N = I − M . By Lemma 16.6, we have kM k = kN k. The corollary follows by noting that " # " # " # i h 0 I 0 −I I −1 N . (I + KP ) I K = −I 0 I 0 P ✷ Corollary 16.8 Let P = M̃ −1 Ñ = N M −1 be, respectively, the normalized left and right coprime factorizations. Then " # h i K . = M −1 (I + KP )−1 I K (I + P K)−1 M̃ −1 ∞ I ∞ Proof. This follows from Corollary 16.7 and the fact that " # h i h I −1 −1 M (I + KP ) = (I + KP )−1 I I K ∞ P K i . ∞ ✷ This corollary says that any H∞ controller for the normalized left coprime factorization is also an H∞ controller for the normalized right coprime factorization. Hence one can work with either factorization. 324 H∞ LOOP SHAPING For future reference, we shall define  " # h  I    (I + P K)−1 I  K bP,K :=      0 P i ∞ !−1 if K stabilizes P otherwise and bopt := sup bP,K . K Then bP,K = bK,P and bopt p = 1 − λmax (Y Q) = r 1− h Ñ M̃ i 2 H . The number bP,K can be related to the classical gain and phase margins as shown in Vinnicombe [1993b]. Theorem 16.9 Let P be a SISO plant and K be a stabilizing controller. Then gain margin ≥ 1 + bP,K 1 − bP,K and phase margin ≥ 2 arcsin(bP,K ). Proof. Note that for SISO system |1 + P (jω)K(jω)| p bP,K ≤ p , 1 + |P (jω)|2 1 + |K(jω)|2 ∀ω. So, at frequencies where k := −P (jω)K(jω) ∈ R+ , bP,K ≤ s |1 − k| (1 + |P |2 )(1 + which implies that k≤ k2 |P |2 ) |1 − k| 1−k  = 1+k , k min (1 + |P |2 )(1 + ) P |P |2 ≤s 1 − bP,K , 1 + bP,K  or k ≥ 2 1 + bP,K 1 − bP,K from which the gain margin result follows. Similarly, at frequencies where P (jω)K(jω) = −ejθ , bP,K ≤ s |1 − ejθ | (1 + |P |2 )(1 + 1 ) |P |2 |2 sin 2θ | |2 sin 2θ | , =  2 1 2 ) min (1 + |P | )(1 + P |P |2 ≤s 16.2. Loop-Shaping Design 325 which implies θ ≥ 2 arcsin bP,K . ✷ For example, bP,K = 1/2 guarantees a gain margin of 3 and a phase margin of 60o . Illustrative MATLAB Commands: ≫ bp,k = emargin(P, K); % given P and K, compute bP,K . ≫ [Kopt , bp,k ] = ncfsyn(P, 1); % find the optimal controller Kopt . ≫ [Ksub , bp,k ] = ncfsyn(P, 2); % find a suboptimal controller Ksub . 16.2 Loop-Shaping Design This section considers the H∞ loop-shaping design. The objective of this approach is to incorporate the simple performance/robustness tradeoff obtained in the loop-shaping with the guaranteed stability properties of H∞ design methods. Recall from Section 6.1 of Chapter 6 that good performance controller design requires that



σ (I + P K)−1 , σ (I + P K)−1 P , σ (I + KP )−1 , σ K(I + P K)−1 (16.1) be made small, particularly in some low-frequency range. Good robustness requires that

σ P K(I + P K)−1 , σ KP (I + KP )−1 (16.2) be made small, particularly in some high-frequency range. These requirements, in turn, imply that good controller design boils down to achieving the desired loop (and controller) gains in the appropriate frequency range: σ(P K) ≫ 1, σ(KP ) ≫ 1, σ(K) ≫ 1 in some low-frequency range and σ(P K) ≪ 1, σ(KP ) ≪ 1, σ(K) ≤ M in some high-frequency range where M is not too large. The H∞ loop-shaping design procedure is developed by McFarlane and Glover [1990, 1992] and is stated next. Loop-Shaping Design Procedure (1) Loop-Shaping: The singular values of the nominal plant, as shown in Figure 16.3, are shaped, using a precompensator W1 and/or a postcompensator W2 , to give a desired open-loop shape. The nominal plant P and the shaping functions W1 , W2 are combined to form the shaped plant, Ps , where Ps = W2 P W1 . We assume that W1 and W2 are such that Ps contains no hidden modes. 326 H∞ LOOP SHAPING r✲e − ✻ ✲ K di ✲❄ e u ✲ P d ✲❄ e y✲ n ❄ e✛ Figure 16.3: Standard feedback configuration (2) Robust Stabilization: a) Calculate ǫmax (i.e., bopt (Ps )), where ǫmax = = inf K stabilizing r 1− h Ñs " I K M̃s # i (I + Ps K)−1 M̃s−1 2 H ∞ !−1 <1 and M̃s , Ñs define the normalized coprime factors of Ps such that Ps = M̃s−1 Ñs and M̃s M̃s∼ + Ñs Ñs∼ = I. If ǫmax ≪ 1 return to (1) and adjust W1 and W2 . b) Select ǫ ≤ ǫmax ; then synthesize a stabilizing controller K∞ that satisfies " # I ≤ ǫ−1 . (I + Ps K∞ )−1 M̃s−1 K∞ ∞ (3) The final feedback controller K is then constructed by combining the H∞ controller K∞ with the shaping functions W1 and W2 , as shown in Figure 16.4, such that K = W1 K∞ W2 . A typical design works as follows: the designer inspects the open-loop singular values of the nominal plant and shapes these by pre- and/or postcompensation until nominal performance (and possibly robust stability) specifications are met. (Recall that the open-loop shape is related to closed-loop objectives.) A feedback controller K∞ with associated stability margin (for the shaped plant) ǫ ≤ ǫmax , is then synthesized. If ǫmax is small, then the specified loop shape is incompatible with robust stability requirements and should be adjusted accordingly; then K∞ is reevaluated. In the preceding design procedure we have specified the desired loop shape by W2 P W1 . But after Stage (2) of the design procedure, the actual loop shape achieved 16.2. Loop-Shaping Design e ✲ K∞ −✻ 327 ✲ W1 ✲ P ✲ W2 ✲ W1 ✲ P ✲ W2 ✲ Ps e ✲ W2 −✻ ✲ K∞ ✲ W1 ✲ P K Figure 16.4: The loop-shaping design procedure is, in fact, given by W1 K∞ W2 P at plant input and P W1 K∞ W2 at plant output. It is therefore possible that the inclusion of K∞ in the open-loop transfer function will cause deterioration in the open-loop shape specified by Ps . In the next section, we will show that the degradation in the loop shape caused by the H∞ controller K∞ is limited at frequencies where the specified loop shape is sufficiently large or sufficiently small. In particular, we show in the next section that ǫ can be interpreted as an indicator of the success of the loop-shaping in addition to providing a robust stability guarantee for the closed-loop systems. A small value of ǫmax (ǫmax ≪ 1) in Stage (2) always indicates incompatibility between the specified loop shape, the nominal plant phase, and robust closed-loop stability. Remark 16.1 Note that, in contrast to the classical loop-shaping approach, the loopshaping here is done without explicit regard for the nominal plant phase information. That is, closed-loop stability requirements are disregarded at this stage. Also, in contrast with conventional H∞ design, the robust stabilization is done without frequency weighting. The design procedure described here is both simple and systematic and only assumes knowledge of elementary loop-shaping principles on the part of the designer. ✸ Remark 16.2 The preceding robust stabilization objective can also be interpreted as the more standard H∞ problem formulation of minimizing the H∞ norm of the frequency-weighted gain from disturbances on the plant input and output to the con- 328 H∞ LOOP SHAPING troller input and output as follows: " # I = (I + Ps K∞ )−1 M̃s−1 K∞ " ∞ " I K∞ # (I + Ps K∞ )−1 # h I Ps i ∞ i W2 −1 −1 (I + P K) W P W 1 2 W1−1 K " # i h I (I + K∞ Ps )−1 I K∞ Ps ∞ # " −1 i h W1 (I + KP )−1 W1 KW2−1 W2 P = = = h ∞ ∞ This shows how all the closed-loop objectives in equations (16.1) and (16.2) are incorporated. As an example, it is easy to see that the signal relationship in Figure 16.5 is given by " # " # " # i w h z1 W2 1 −1 −1 . = (I + P K) W2 P W1 w2 z2 W1−1 K ✸ z2 ✻ W1−1 e ✲ K − ✻ ✻ w2 ❄ W1 ❄✲ ✲e P w1 ❄ W2−1 ❄✲ ✲e W2 z1 ✲ Figure 16.5: An equivalent H∞ formulation 16.3 Justification for H∞ Loop Shaping The objective of this section is to provide justification for the use of parameter ǫ as a design indicator. We will show that ǫ is a measure of both closed-loop robust stability and the success of the design in meeting the loop-shaping specifications. Readers are encouraged to consult the original reference by McFarlane and Glover [1990] for further details. We first examine the possibility of loop shape deterioration at frequencies of high loop gain (typically low-frequency). At low-frequency [in particular, ω ∈ (0, ωl )], the deterioration in loop shape at plant output can be obtained by comparing σ(P W1 K∞ W2 ) 16.3. Justification for H∞ Loop Shaping 329 to σ(Ps ) = σ(W2 P W1 ). Note that σ(P K) = σ(P W1 K∞ W2 ) = σ(W2−1 W2 P W1 K∞ W2 ) ≥ σ(W2 P W1 )σ(K∞ )/κ(W2 ) (16.3) where κ(·) denotes condition number. Similarly, for loop shape deterioration at plant input, we have σ(KP ) = σ(W1 K∞ W2 P ) = σ(W1 K∞ W2 P W1 W1−1 ) ≥ σ(W2 P W1 )σ(K∞ )/κ(W1 ). (16.4) In each case, σ(K∞ ) is required to obtain a bound on the deterioration in the loop shape at low-frequency. Note that the condition numbers κ(W1 ) and κ(W2 ) are selected by the designer. Next, recalling that Ps denotes the shaped plant and that K∞ robustly stabilizes the normalized coprime factorization of Ps with stability margin ǫ, we have " # I ≤ ǫ−1 := γ (16.5) (I + Ps K∞ )−1 M̃s−1 K∞ ∞ where (Ñs , M̃s ) is a normalized left coprime factorization of Ps , and the parameter γ is defined to simplify the notation to follow. The following result shows that σ(K∞ ) is explicitly bounded by functions of ǫ and σ(Ps ), the minimum singular value of the shaped plant, and hence by equation (16.3) and (16.4) K∞ will only have a limited effect on the specified loop shape at low-frequency. Theorem 16.10 Any controller K∞ satisfying equation (16.5), where Ps is assumed square, also satisfies p σ(Ps (jω)) − γ 2 − 1 σ(K∞ (jω)) ≥ p γ 2 − 1σ(Ps (jω)) + 1 for all ω such that σ(Ps (jω)) > p γ 2 − 1. p p Furthermore, if σ(Ps ) ≫ γ 2 − 1, then σ(K∞ (jω)) ' 1/ γ 2 − 1, where ' denotes asymptotically greater than or equal to as σ(Ps ) → ∞. Proof. First note that σ(Ps ) > p γ 2 − 1 implies that I + Ps Ps∗ > γ 2 I. Further, since (Ñs , M̃s ) is a normalized left coprime factorization of Ps , we have M̃s M̃s∗ = I − Ñs Ñs∗ = I − M̃s Ps Ps∗ M̃s∗ . Then M̃s∗ M̃s = (I + Ps Ps∗ )−1 < γ −2 I. 330 H∞ LOOP SHAPING Now " I K∞ # (I + Ps K∞ )−1 M̃s−1 ∞ ≤γ can be rewritten as ∗ ∗ (I + K∞ K∞ ) ≤ γ 2 (I + K∞ Ps∗ )(M̃s∗ M̃s )(I + Ps K∞ ). (16.6) We will next show that K∞ is invertible. Suppose that there exists an x such that K∞ x = 0, then x∗ × equation (16.6) × x gives γ −2 x∗ x ≤ x∗ M̃s∗ M̃s x, which implies that x = 0 since M̃s∗ M̃s < γ −2 I, and hence K∞ is invertible. Equation (16.6) can now be written as −∗ −1 −∗ −1 (K∞ K∞ + I) ≤ γ 2 (K∞ + Ps∗ )M̃s∗ M̃s (K∞ + Ps ). (16.7) Define W such that (W W ∗ )−1 = I − γ 2 M̃s∗ M̃s = I − γ 2 (I + Ps Ps∗ )−1 . −1 Completing the square in equation (16.7) with respect to K∞ yields −∗ −1 (K∞ + N ∗ )(W W ∗ )−1 (K∞ + N ) ≤ (γ 2 − 1)R∗ R where N ∗ = γ 2 Ps ((1 − γ 2 )I + Ps∗ Ps )−1 R R = (I + Ps∗ Ps )((1 − γ 2 )I + Ps∗ Ps )−1 . Hence we have −∗ −1 R−∗ (K∞ + N ∗ )(W W ∗ )−1 (K∞ + N )R−1 ≤ (γ 2 − 1)I and
p −1 σ W −1 (K∞ + N )R−1 ≤ γ 2 − 1.
−1 −1 Use σ W −1 (K∞ + N )R−1 ≥ σ(W −1 )σ(K∞ + N )σ(R−1 ) to get −1 σ(K∞ + N) ≤ p γ 2 − 1σ(W )σ(R) −1 and use σ(K∞ + N ) ≥ σ(K∞ ) − σ(N ) to get n o−1 σ(K∞ ) ≥ (γ 2 − 1)1/2 σ(W )σ(R) + σ(N ) . (16.8) 16.3. Justification for H∞ Loop Shaping 331 Next, note that the eigenvalues of W W ∗ , N ∗ N , and R∗ R can be computed as follows: λ(W W ∗ ) = λ(N ∗ N ) = λ(R∗ R) = Therefore, 1 + λ(Ps Ps∗ ) 1 − γ 2 + λ(Ps Ps∗ ) γ 4 λ(Ps Ps∗ ) (1 − γ 2 + λ(Ps Ps∗ ))2 1 + λ(Ps Ps∗ ) . 1 − γ 2 + λ(Ps Ps∗ ) 1/2  1/2 1 + λmin (Ps Ps∗ ) 1 + σ2 (Ps ) = 1 − γ 2 + λmin (Ps Ps∗ ) 1 − γ 2 + σ 2 (Ps ) p 2 p γ λmin (Ps Ps∗ ) γ 2 σ(Ps ) σ(N ) = λmax (N ∗ N ) = = 2 ∗ 1 − γ + λmin (Ps Ps ) 1 − γ 2 + σ2 (Ps ) 1/2  1/2  p 1 + λmin (Ps Ps∗ ) 1 + σ 2 (Ps ) ∗ . σ(R) = λmax (R R) = = 1 − γ 2 + λmin (Ps Ps∗ ) 1 − γ 2 + σ 2 (Ps ) p σ(W ) = λmax (W W ∗ ) =  Substituting these formulas into equation (16.8), we have σ(K∞ ) ≥  (γ 2 − 1)1/2 (1 + σ 2 (Ps )) + γ 2 σ(Ps ) σ2 (Ps ) − (γ 2 − 1) −1 p σ(Ps )) − γ 2 − 1 =p . γ 2 − 1σ(Ps ) + 1 ✷ The main implication of Theorem 16.10 is that the bound on σ(K∞ ) depends only on the selected loop shape and the stability margin of the shaped plant. The value of γ(= ǫ−1 ) directly determines the frequency range over which this result is valid – a small γ (large ǫ) is desirable, as we would expect. Further, Ps has a sufficiently large loop gain; then so also will Ps K∞ provided that γ(= ǫ−1 ) is sufficiently small. In an analogous manner, we now examine the possibility of deterioration in the loop shape at high-frequency due to the inclusion of K∞ . Note that at high frequency [in particular, ω ∈ (ωh , ∞)] the deterioration in plant output loop shape can be obtained by comparing σ(P W1 K∞ W2 ) to σ(Ps ) = σ(W2 P W1 ). Note that, analogous to equation (16.3) and (16.4), we have σ(P K) = σ(P W1 K∞ W2 ) ≤ σ(W2 P W1 )σ(K∞ )κ(W2 ). Similarly, the corresponding deterioration in plant input loop shape is obtained by comparing σ(W1 K∞ W2 P ) to σ(W2 P W1 ), where σ(KP ) = σ(W1 K∞ W2 P ) ≤ σ(W2 P W1 )σ(K∞ )κ(W1 ). 332 H∞ LOOP SHAPING Hence, in each case, σ(K∞ ) is required to obtain a bound on the deterioration in the loop shape at high-frequency. In an identical manner to Theorem 16.10, we now show that σ(K∞ ) is explicitly bounded by functions of γ and σ(Ps ), the maximum singular value of the shaped plant. Theorem 16.11 Any controller K∞ satisfying equation (16.5) also satisfies p γ 2 − 1 + σ(Ps (jω)) p σ(K∞ (jω)) ≤ 1 − γ 2 − 1 σ(Ps (jω)) for all ω such that 1 . σ(Ps (jω)) < p 2 γ −1 p p Furthermore, if σ(Ps ) ≪ 1/ γ 2 − 1, then σ(K∞ (jω)) / γ 2 − 1, where / denotes asymptotically less than or equal to as σ(Ps ) → 0. Proof. The proof of Theorem 16.11 is similar to that of Theorem 16.10 and is only sketched here: As in the proof of Theorem 16.10, we have M̃s∗ M̃s = (I + Ps Ps∗ )−1 and ∗ ∗ (I + K∞ K∞ ) ≤ γ 2 (I + K∞ Ps∗ )(M̃s∗ M̃s )(I + Ps K∞ ). 1 , γ 2 −1 (16.9) Since σ(Ps ) < √ I − γ 2 Ps∗ (I + Ps Ps∗ )−1 Ps > 0 and there exists a spectral factorization V ∗ V = I − γ 2 Ps∗ (I + Ps Ps∗ )−1 Ps . Now, completing the square in equation (16.9) with respect to K∞ yields ∗ (K∞ + M ∗ )V ∗ V (K∞ + M ) ≤ (γ 2 − 1)Y ∗ Y where M Y ∗Y Hence we have which implies = γ 2 Ps∗ (I + (1 − γ 2 )Ps Ps∗ )−1 = (γ 2 − 1)(I + Ps Ps∗ )(I + (1 − γ 2 )Ps Ps∗ )−1 .
p σ V (K∞ + M )Y −1 ≤ γ 2 − 1, p γ2 − 1 σ(K∞ ) ≤ + σ(M ). σ(V )σ(Y −1 ) (16.10) 16.3. Justification for H∞ Loop Shaping 333 As in the proof of Theorem 16.10, it is easy to show that σ(V ) = σ(Y −1 ) = σ(M ) =  1 − (γ 2 − 1)σ2 (Ps ) 1 + σ2 (Ps ) 1/2 γ 2 σ(Ps ) . 1 − (γ 2 − 1)σ 2 (Ps ) Substituting these formulas into equation (16.10), we have p γ 2 − 1 + σ(Ps ) (γ 2 − 1)1/2 (1 + σ2 (Ps )) + γ 2 σ(Ps ) p σ(K∞ ) ≤ = . 2 1 − (γ 2 − 1)σ (Ps ) 1 − γ 2 − 1σ(Ps ) ✷ The results in Theorems 16.10 and 16.11 confirm that γ (alternatively ǫ) indicates the compatibility between the specified loop shape and closed-loop stability requirements. Theorem 16.12 Let P be the nominal plant and let K = W1 K∞ W2 be the associated controller obtained from loop-shaping design procedure in the last section. Then if " # I ≤γ (I + Ps K∞ )−1 M̃s−1 K∞ ∞ we have σ K(I + P K)−1
≤ γσ(M̃s )σ(W1 )σ(W2 ) n o
σ (I + P K)−1 ≤ min γσ(M̃s )κ(W2 ), 1 + γσ(Ns )κ(W2 ) n o
σ K(I + P K)−1 P ≤ min γσ(Ñs )κ(W1 ), 1 + γσ(Ms )κ(W1 ) γσ(Ñs ) σ(W1 )σ(W2 ) n o
−1 ≤ min 1 + γσ(Ñs )κ(W1 ), γσ(Ms )κ(W1 ) σ (I + KP ) n o
σ G(I + KP )−1 K ≤ min 1 + γσ(M̃s )κ(W2 ), γσ(Ns )κ(W2 ) σ (I + P K)−1 P where
≤ 1/2 σ2 (W2 P W1 ) 1 + σ2 (W2 P W1 ) 1/2  1 σ(M̃s ) = σ(Ms ) = 1 + σ2 (W2 P W1 ) σ(Ñs ) = σ(Ns ) =  (16.11) (16.12) (16.13) (16.14) (16.15) (16.16) (16.17) (16.18) and (Ñs , M̃s ), (Ns , Ms ) is a normalized left coprime factorization and right coprime factorization, respectively, of Ps = W2 P W1 . 334 H∞ LOOP SHAPING Proof. Note that M̃s∗ M̃s = (I + Ps Ps∗ )−1 and M̃s M̃s∗ = I − Ñs Ñs∗ . Then 1 σ 2 (M̃s ) = λmax (M̃s∗ M̃s ) = σ2 (Ñs ) = 1 − = 1 + λmin (Ps Ps∗ ) σ 2 (Ps ) σ2 (M̃s ) = 2 1 1 + σ2 (Ps ) . 1 + σ (Ps ) The proof for the normalized right coprime factorization is similar. All other inequalities follow from noting that " # I ≤γ (I + Ps K∞ )−1 M̃s−1 K∞ ∞ and " I K∞ # −1 (I + Ps K∞ ) M̃s−1 = ∞ = " W2 W1−1 K " # W1−1 W2 P # (I + P K)−1 (I + KP )−1 h h W2−1 W1 P W1 P W2−1 i i ∞ ∞ ✷ This theorem shows that all closed-loop objectives are guaranteed to have bounded magnitude and the bounds depend only on γ, W1 , W2 , and P . 16.4 Further Guidelines for Loop Shaping Let P = N M −1 be a normalized right coprime factorization. It was shown in Georgiou and Smith [1990] that #! " M (s) bopt (P ) ≤ λ(P ) := inf σ . Res>0 N (s) Hence a small λ(P ) would necessarily imply a small bopt (P ). We shall now discuss the performance limitations implied by this relationship for a scalar system. The following argument is based on Vinnicombe [1993b], to which the reader is referred for further discussions. Let z1 , z2 , . . . , zm and p1 , p2 , . . . , pk be the open right-half plane zeros and poles of the plant P . Define Nz (s) = zm − s z1 − s z2 − s ... , z1 + s z2 + s zm + s Np (s) = p1 − s p2 − s pk − s ... . p1 + s p2 + s pk + s 16.4. Further Guidelines for Loop Shaping 335 Then P can be written as P (s) = P0 (s)Nz (s)/Np (s) where P0 (s) has no open right-half plane poles or zeros. Let N0 (s) and M0 (s) be stable and minimum phase spectral factors: N0 (s)N0∼ (s) =  1 1+ P (s)P ∼ (s) −1 , M0 (s)M0∼ (s) = (1 + P (s)P ∼ (s))−1 . Then P0 = N0 /M0 is a normalized coprime factorization and (N0 Nz ) and (M0 Np ) form a pair of normalized coprime factorizations of P . Thus q bopt (P ) ≤ |N0 (s)Nz (s)|2 + |M0 (s)Np (s)|2 , ∀Re(s) > 0. (16.19) Since N0 and M0 are both stable and have no zeros in the open right-half plane, ln(N0 (s)) and ln(M0 (s)) are both analytic in Re(s) > 0 and so can be determined from their boundary values on Re(s) = 0 via Poisson integrals (see also Problem 16.15): ln |N0 (re )| = Z ∞ ln |M0 (rejθ )| = Z ∞ jθ −∞ where r > 0, −π/2 < θ < π/2, and Kθ (ω/r) ln −∞ = ln ! 1 p Kθ (ω/r) d(ln ω) 1 + 1/|P (jω)|2 ! 1 p Kθ (ω/r) d(ln ω) 1 + |P (jω)|2 1 2(ω/r)[1 + (ω/r)2 ] cos θ π [1 − (ω/r)2 ]2 + 4(ω/r)2 cos2 θ The function Kθ (ω/r) is plotted in Figure 16.6 against logarithmic frequency for various values of θ. Note that the function is symmetric to ω = r in log ω and it attends the maximum at ω = r. The function converges to an impulse function at ω = r when θ approaches 90o ; that is, when |N0 (s)| or |M0 (s)| is evaluated close to the imaginary axis. Since the kernel function Kθ (ω/r) has the greatest weighting near ω = r, the Poisson integral is largely determined by the frequency response near that frequency. Thus it is clear that |N0 (rejθ )| will be small if |P (jω)| is small near ω = r. Similarly, |M0 (rejθ )| will be small if |P (jω)| is large near ω = r. It is also important to note that a very large percentage of weighting is concentrated in a very narrow frequency range for a large θ (i.e., when s = rejθ has a much larger imaginary part than the real part). Hence |N0 (rejθ )| and |M0 (rejθ )| will essentially be determined by |P (jω)| in a very narrow frequency range near ω = r when θ is large. On the other hand, when θ is small, a larger range of frequency response |P (jω)| around ω = r will have affect on the value |N0 (rejθ )| and |M0 (rejθ )|. (This, in fact, will imply 336 H∞ LOOP SHAPING 2 θ = 80 1 θ = 60 Kθ (ω/r) 1.5 o o o θ = 30 0.5 θ =0 0 −2 10 −1 10 ω/r 0 10 o 1 10 2 10 Figure 16.6: Kθ (ω/r) vs. normalized frequency ω/r that a right-plane zero (pole) with a much larger real part than the imaginary part will have much worse effect on the performance than a right-plane zero (pole) with a much larger imaginary part than the real part.) Let s = rejθ . Consider again the bound of equation (16.19) and note that Nz (zi ) = 0 and Np (pj ) = 0, we see that there are several ways in which the bound may be small (i.e., bopt (P ) is small). ✄ |Nz (s)| and |Np (s)| are both small for some s. That is, |Nz (s)| ≈ 0 (i.e., s is close to a right-half plane zero of P ) and |Np (s)| ≈ 0 (i.e., s is close to a right-half plane pole of P ). This is only possible if P (s) has a right-half plane zero near a right-half plane pole. (See Example 16.1.) ✄ |Nz (s)| and |M0 (s)| are both small for some s. That is, |Nz (s)| ≈ 0 (i.e., s is close to a right-half plane zero of P ) and |M0 (s)| ≈ 0 (i.e., |P (jω)| is large around ω = |s| = r). This is only possible if |P (jω)| is large around ω = r, where r is the modulus of a right-half plane zero of P . (See Example 16.2.) ✄ |Np (s)| and |N0 (s)| are both small for some s. That is, |Np (s)| ≈ 0 (i.e., s is close to a right-half plane pole of P ) and |N0 (s)| ≈ 0 (i.e., |P (jω)| is small around ω = |s| = r). This is only possible if |P (jω)| is small around ω = r, where r is the modulus of a right-half plane pole of P . (See Example 16.3.) ✄ |N0 (s)| and |M0 (s)| are both small for some s. That is, |N0 (s)| ≈ 0 (i.e., |P (jω)| is small around ω = |s| = r) and |M0 (s)| ≈ 0 (i.e., |P (jω)| is large around ω = |s| = r). The only way in which |P (jω)| can be both small and large 16.4. Further Guidelines for Loop Shaping 337 at frequencies near ω = r is that |P (jω)| is approximately equal to 1 and the absolute value of the slope of |P (jω)| is large. (See Example 16.4.) Example 16.1 Consider an unstable and nonminimum phase system P1 (s) = K(s − r) . (s + 1)(s − 1) The frequency responses of P1 (s) with r = 0.9 and K = 0.1, 1, and 10 are shown in Figure 16.7. The following table shows that bopt (P1 ) will be very small for all K whenever r is close to 1 (i.e., whenever there is an unstable pole close to an unstable zero). K = 0.1 r bopt (P1 ) 0.5 0.0125 0.7 0.0075 0.9 0.0025 1.1 0.0025 1.3 0.0074 1.5 0.0124 K=1 r bopt (P1 ) 0.5 0.1036 0.7 0.0579 0.9 0.0179 1.1 0.0165 1.3 0.0457 1.5 0.0706 K = 10 r bopt (P1 ) 0.5 0.0658 0.7 0.0312 0.9 0.0088 1.1 0.0077 1.3 0.0208 1.5 0.0318 1 10 K=10 0 10 K=1 −1 10 K=0.1 −2 10 −2 10 −1 10 0 10 1 10 2 10 Figure 16.7: Frequency responses of P1 for r = 0.9 and K = 0.1, 1, and 10 338 H∞ LOOP SHAPING Example 16.2 Consider a nonminimum phase plant P2 (s) = K(s − 1) . s(s + 1) The frequency responses of P2 (s) with K = 0.1, 1, and 10 are shown in Figure 16.8. The following table shows clearly that bopt (P2 ) will be small if |P2 (jω)| is large around ω = 1, the modulus of the right-half plane zero. K 0.01 0.1 1 10 100 bopt (P2 ) 0.7001 0.6451 0.3827 0.0841 0.0098 3 10 K=10 2 10 1 10 K=1 0 10 K=0.1 −1 10 −2 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 Figure 16.8: Frequency responses of P2 for K = 0.1, 1, and 10 Note that bopt (L/s) = 0.707 for any L and bopt (P2 ) −→ 0.707 as K −→ 0. This is because |P2 (jω)| around the frequency of the right-half plane zero is very small as K −→ 0. Next consider a plant with a pair of complex right-half plane zeros: P3 (s) = K[(s − cos θ)2 + sin2 θ] . s[(s + cos θ)2 + sin2 θ] The magnitude frequency response of P3 is the same as that of P2 for the same K. The optimal bopt (P3 ) for various θ’s are listed in the following table: 16.4. Further Guidelines for Loop Shaping 339 K = 0.1 θ (degree) bopt (P3 ) 0 0.5952 45 0.6230 60 0.6447 80 0.6835 85 0.6950 K =1 θ (degree) bopt (P3 ) 0 0.2588 45 0.3078 60 0.3568 80 0.4881 85 0.5512 K = 10 θ (degree) bopt (P3 ) 0 0.0391 45 0.0488 60 0.0584 80 0.0813 85 0.0897 It can also be concluded from the table that bopt (P3 ) will be small if |P3 (jω)| is large around the frequency of ω = 1 (the modulus of the right-half plane zero). It can be further concluded that, for zeros with the same modulus, bopt (P3 ) will be smaller for a plant with relatively larger real part zeros than for a plant with relatively larger imaginary part zeros (i.e., a pair of real right-half plane zeros has a much worse effect on the performance than a pair of almost pure imaginary axis right-half plane zeros of the same modulus). Example 16.3 Consider an unstable plant P4 (s) = K(s + 1) . s(s − 1) The magnitude frequency response of P4 is again the same as that of P2 for the same K. The following table shows that bopt (P4 ) will be small if |P4 (jω)| is small around ω = 1 (the modulus of the right-half plane pole). K 0.01 0.1 1 10 100 bopt (P4 ) 0.0098 0.0841 0.3827 0.6451 0.7001 Note that bopt (P4 ) −→ 0.707 as K −→ ∞. This is because |P4 (jω)| is very large around the frequency of the modulus of the right-half plane pole as K −→ ∞. Next consider a plant with complex right-half plane poles: P5 (s) = K[(s + cos θ)2 + sin2 θ] . s[(s − cos θ)2 + sin2 θ] The optimal bopt (P5 ) for various θ’s are listed in the following table: 340 H∞ LOOP SHAPING K = 0.1 θ (degree) bopt (P5 ) 0 0.0391 45 0.0488 60 0.0584 80 0.0813 85 0.0897 K =1 θ (degree) bopt (P5 ) 0 0.2588 45 0.3078 60 0.3568 80 0.4881 85 0.5512 K = 10 θ (degree) bopt (P5 ) 0 0.5952 45 0.6230 60 0.6447 80 0.6835 85 0.6950 It can also be concluded from the table that bopt (P5 ) will be small if |P5 (jω)| is small around the frequency of the modulus of the right-half plane pole. It can be further concluded that, for poles with the same modulus, bopt (P5 ) will be smaller for a plant with relatively larger real part poles than for a plant with relatively larger imaginary part poles (i.e., a pair of real right-half plane poles has a much worse effect on the performance than a pair of almost pure imaginary axis right-half plane poles of the same modulus). Example 16.4 Let a stable and minimum phase transfer function be P6 (s) = K(0.2s + 1)4 . s(s + 1)4 15 10 K=10000 10 10 5 K=0.1 10 0 10 K=0.00001 −5 10 −10 10 −15 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 Figure 16.9: Frequency response of P6 for K = 10−5 , 10−1 and 104 16.5. Notes and References 341 The frequency responses of P6 with K = 10−5 , 10−1 , and 104 are shown in Figure 16.9. It is clear that the slope of the frequency response near the crossover frequency for K = 10−5 is not too large, which implies a reasonably good loop shape. Thus we should expect bopt (P6 ) to be not too small. A similar conclusion applies to K = 104 . On the other hand, the slope of the frequency response near the crossover frequency for K = 0.1 is quite large which implies a bad loop shape. Thus we should expect bopt (P6 ) to be quite small. This is clear from the following table: K 10−5 10−3 0.1 1 10 102 104 bopt (P6 ) 0.3566 0.0938 0.0569 0.0597 0.0765 0.1226 0.4933 Based on the preceding discussion, we can give some guidelines for the loop-shaping design. ♥ The loop transfer function should be shaped in such a way that it has low gain around the frequency of the modulus of any right-half plane zero z. Typically, it requires that the crossover frequency be much smaller than the modulus of the right-half plane zero; say, ωc < |z|/2 for any real zero and ωc < |z| for any complex zero with a much larger imaginary part than the real part (see Figure 16.6). ♥ The loop transfer function should have a large gain around the frequency of the modulus of any right-half plane pole. ♥ The loop transfer function should not have a large slope near the crossover frequencies. These guidelines are consistent with the rules used in classical control theory (see Bode [1945] and Horowitz [1963]). 16.5 Notes and References The H∞ loop-shaping using normalized coprime factorization was developed by McFarlane and Glover [1990, 1992], on which most parts of this chapter is based. In the same references, some design examples were also shown. The method has been applied to the design of scheduled controllers for a VSTOL aircraft in Hyde and Glover [1993]. Some limitations of this loop-shaping design are discussed in detail in Vinnicombe [1993b] (on which Section 16.4 is based) and Christian and Freudenberg [1994]. The robust stabilization of normalized coprime factors is closely related to the robustness in the gap metric and ν-gap metric, which will be discussed in the next chapter, see El-Sakkary [1985], Georgiou and Smith [1990], Glover and McFarlane [1989], McFarlane, Glover, and Vidyasagar [1990], Qiu and Davison [1992a, 1992b], Vinnicombe [1993a, 1993b], Vidyasagar [1984, 1985], Zhu [1989], and references therein. 342 16.6 H∞ LOOP SHAPING Problems 1 . Compute by hand ǫmax , s−2 the maximum stability radius for a normalized coprime factor perturbation of G(s). Problem 16.1 Consider a feedback system with G(s) = Problem 16.2 In Corollary 16.2, find the parameterization of all H∞ controllers. Problem 16.3 Let P∆ = (N + ∆N )(M + ∆M )−1 be a right coprime factor perturbed plant with a nominal plant P = N M −1 , where (N, M ) is a pair of normalized right coprime factorization. Formulate the corresponding robust stabilization problem as an H∞ control problem and find a stabilizing controller using the H∞ formulas in Chapter 14. Are there any connections between this stabilizing controller and the controller obtained in this chapter for left coprime stabilization? Problem 16.4 Let P have coprime factorizations P = N M −1 = M̃ −1 Ñ . Then there exist U, V, Ũ , Ṽ ∈ H∞ such that " #" # Ṽ −Ũ M U = I. N V Ñ M̃ Furthermore, all stabilizing controllers for P can be written as K = (U + M Q)(V + N Q)−1 , Show that " K I # (I + P K)−1 M̃ −1 = " Q ∈ H∞ . U + MQ V + NQ # . Suppose P = N M −1 = M̃ −1 Ñ are normalized coprime factorizations. Show that " # " # " # R+Q U M −1 = + Q bP,K = I V N ∞ ∞ where R = M ∼ U + N ∼ V . Problem 16.5 Let K be a controller that stabilizes the plant P . Show that 1. K stabilizes P̃ = P + ∆a such that ∆a ∈ H∞ and k∆a k < bP,K ; 2. K stabilizes P̃ = P (I + ∆m ) such that ∆m ∈ H∞ and k∆m k < bP,K ; 3. K stabilizes P̃ = (I + ∆m )P such that ∆m ∈ H∞ and k∆m k < bP,K ; 4. K stabilizes P̃ = P (I + ∆f )−1 such that ∆f ∈ H∞ and k∆f k < bP,K ; 5. K stabilizes P̃ = (I + ∆f )−1 P such that ∆f ∈ H∞ and k∆f k < bP,K . 16.6. Problems 343 Discuss the possible implications of the preceding results. Problem 16.6 Let K be a controller that stabilizes the plant P . Show that −1 1. any controller " # in the form of K̃ = (U + ∆U )(V + ∆V ) such that ∆U , ∆V ∈ H∞ ∆U and < bP,K also stabilizes P ; ∆V ∞ 2. any controller K̃ = K + ∆a such that ∆a ∈ H∞ and k∆a k < bP,K also stabilizes P; 3. any controller K̃ = K(I + ∆m ) such that ∆m ∈ H∞ and k∆m k < bP,K also stabilizes P ; 4. any controller K̃ = (I + ∆m )K such that ∆m ∈ H∞ and k∆m k < bP,K also stabilizes P ; 5. any controller K̃ = K(I + ∆f )−1 such that ∆f ∈ H∞ and k∆f k < bP,K also stabilizes P ; 6. any controller K̃ = (I + ∆f )−1 K such that ∆f ∈ H∞ and k∆f k < bP,K also stabilizes P . Discuss the possible implications of the preceding results. Problem 16.7 Let an uncertain plant be given by Pδ = s+α , α ∈ [1, 3], ζ ∈ [0.2, 0.4] s2 + 2ζs + 1 and let a nominal model be P = s2 s + α0 . + 2ζ0 s + 1 1. Let α0 = 2 and ζ0 = 0.3. Find the largest possible k∆add k∞ and k∆mul k∞ where ∆add = Pδ − P, ∆mul = (Pδ − P )/P. 2. Let α0 = 2 and ζ0 = 0.3. Show that P = N/M with N= s2 s2 + 0.6s + 1 s+2 , M= 2 + 1.9576s + 2.2361 s + 1.9576s + 2.2361 is a normalized coprime factorization. Now let Nδ = s2 + 2ζs + 1 s+α , M = δ s2 + 1.9576s + 2.2361 s2 + 1.9576s + 2.2361 ∆n = Nδ − N, ∆m = Mδ − M. i h . Find the largest possible ∆n ∆m ∞ 344 H∞ LOOP SHAPING 3. In part 2, let (Nhδ , Mδ ) be a inormalized coprime factorization of Pδ . Find the . largest possible ∆n ∆m ∞ 4. Find the optimal h nominal αi0 and ζ0 such that the largest possible k∆add k∞ , are minimized, respectively. k∆mul k∞ , and ∆n ∆m ∞ Discuss the advantages of each uncertainty modeling method in terms of robust stabilizations. −10 . Design (a) a precompensator W of order no greater s(s − 1) than 2 such that the crossover frequency ωc ≤ 2 and bopt (W P ) is as large as possible; (b) find the optimal loop-shaping controller K = K∞ W with the W obtained in part (a). Problem 16.8 Let P = 100(1 − s) . Design (a) a precompensator W of order no s(s + 10) greater than 2 such that the crossover frequency ωc ≥ 1 and bopt (W P ) is as large as possible; (b) find the optimal loop-shaping controller K = K∞ W with the W obtained in part (a). # " A B and G(s) = N M −1 with Problem 16.10 Let G(s) = C 0 Problem 16.9 Let P = " N M #   = A + BF C F B   0  I −1 where is a normalized right Let # #F is chosen such that N M " # coprime factorization. " " N̂ N̂ N is also a be an rth order balanced truncation of . Show that M̂ M̂ M normalized right coprime factorization. Problem 16.11 (Reduced-Order Controllers by Controller Model" Reduction; # see McA B Farlane and Glover [1990], Zhou and Chen [1995].) Let G(s) = = M̃ −1 Ñ C 0 be a normalized left coprime factorization and let K(s) be a suboptimal controller given in Corollary 18.2 (with performance γ). Let K = U V −1 be a right coprime factorization   " # A − BB ∗ X∞ −Y C ∗ U   = −C I  V ∗ 0 −B X∞ 16.6. Problems 345 and Û , V̂ ∈ RH∞ be approximations of U and V . Define " # " # Û U − ǫ := V V̂ ∞ and Kr = Û V̂ −1 . Show that Kr is a stabilizing controller for G if ǫ < 1 and " " # # h i Kr Kr γ −1 −1 . < = (I + GKr )−1 I G (I + GKr ) M̃ 1−ǫ I I ∞ ∞ Problem 16.12 (Reduced Order Controllers by Plant Model Reduction; see McFarlane and Glover [1990].) Let G = M̃ −1 Ñ be a normalized left coprime factorization and let K be a stabilizing controller such that # " K (I + GK)−1 M̃ −1 ≤ δ −1 . I ∞ Let Gr := M̃r−1 Ñr be an approximation of G and i h ǫ := M̃ − M̃r Ñ − Ñr (a) Show that K stabilizes Gr and " # K (I + Gr K)−1 M̃r−1 I ∞ < δ. ≤ (δ − ǫ)−1 . ∞ (b) Let W (s), W −1 (s) ∈ RH∞ be obtained from the following spectral factorization: W −1 W −∗ = M̃r M̃r∗ + Ñr Ñr∗ . Show that kW k∞ ≤ (c) Show that = inf K1 and " 1 and W −1 1−ǫ −1 δrn := inf K1 I # " K1 " K1 I # (I + Gr K1 )−1 K1 I # ∞ ≤ 1 + ǫ. (I + Gr K1 )−1 (W M̃r )−1 ∞ h I Gr i (I + Gr K1 )−1 M̃r−1 ∞ ∞ ≤ W −1 ∞ 1+ǫ ≤ . δ−ǫ δ−ǫ −1 ≤ δrn kW k∞ . 346 H∞ LOOP SHAPING (d) With the controller K1 given in (c), show that # " # " h K K1 1 (I + GK1 )−1 I = (I + GK1 )−1 M̃ −1 I I G ∞ i ∞ −1 ≤ δred where δred := δrn δ−ǫ 1−ǫ −ǫ≤ −ǫ≤ (δ − ǫ) − ǫ. kW k∞ kW −1 k∞ kW k∞ 1+ǫ Note that ifpÑr and M̃r are the kth-order balanced truncation of Ñ and M̃ , then Pn δ = δrn = 1 − σ12 , δred = δ − ǫ, and ǫ ≤ 2 i=k+1 σi , where σi are the Hankel h i singular values of M̃ Ñ . (e) Show that δ̃r−1 := inf K2 " K2 I # (I + Gr K2 )−1 M̃r−1 ∞ ≤ (δ − ǫ)−1 . (f) With the controller K2 given in (e), show that " # " # h K2 K2 −1 −1 (I + GK2 ) M̃ (I + GK2 )−1 I = I I ∞ G i ∞ ≤ (δ̃r − ǫ)−1 ≤ (δ − 2ǫ)−1 . Again note that if Ñr and M̃r are the kth-order balanced truncation of Ñ and M̃ , then δ̃r = δ. (Note that K1 and K2 are reduced-order controllers.) " # A B Problem 16.13 Let G(s) = = M̃ −1 Ñ be a normalized left coprime factorC 0 ization and let K(s) be a suboptimal controller given in Corollary 18.2 (with performance γ): " # A − BB ∗ X∞ − Y C ∗ C −Y C ∗ K(s) = −B ∗ X∞ 0 where X∞ = and  −1 γ2 γ2 Q I − Y Q γ2 − 1 γ2 − 1 AY + Y A∗ − Y C ∗ CY + BB ∗ = 0 16.6. Problems 347 Q(A − Y C ∗ C) + (A − Y C ∗ C)∗ Q + C ∗ C = 0. Suppose Y and Q are balanced; that is, Y = Q = diag(σ1 , . . . , σr , σr+1 , . . . , σn ) = diag(Σ1 , Σ2 ) and let G(s) be partitioned accordingly as  A11  G(s) =  A21 C1 Denote Y1 = Σ1 and X1 = Kr (s) = γ2 γ 2 −1 Σ1 "  I−  B1  B2  . 0 A12 A22 C2 γ2 2 γ 2 −1 Σ1 −1 . Show that A11 − B1 B1∗ X1 − Y1 C1∗ C1 −B1∗ X1 −Y1 C1∗ 0 # is exactly the reduced-order controller obtained from the last problem with balanced model reduction procedure. (It is also interesting to note that Q = X(I + Y X)−1 where X = X ∗ ≥ 0 is the stabilizing solution to XA + A∗ X − XBB ∗ X + C ∗ C = 0. Hence balancing Y and Q is equivalent to balancing X and Y . This is called Riccati balancing; see Jonckheere and Silverman [1983].) Problem 16.14 Apply the controller reduction methods in the last three problems, # " A B where respectively, to a satellite system G(s) = C 0    A=   0 0 0 0 1 0 0 0 0 0 0 −1.5392 0 0 1 −2 × 0.003 × 1.539 C= h 1 0 1 0 i ,    ,      B=   0 1.7319 × 10−5 0 3.7859 × 10−4 D = 0. Compare the results (see McFarlane and Glover [1990] for further details).    ,   348 H∞ LOOP SHAPING Problem 16.15 Let f (s) be analytic in the closed right-half plane and suppose |f (rejθ ) = 0. r→∞ θ∈[−π/2,π/2] r lim max Then the Poisson integral formula (see, for example, Freudenberg and Looze [1988], page 37) says that f (s) at any point s = x + jy in the open right-half plane can be recovered from f (jω) via the integral relation: Z x 1 ∞ dω. f (jω) 2 f (s) = π −∞ x + (y − ω)2 Let s = rejθ (i.e., x = r cos θ and y = r sin θ) with r > 0 and −π/2 < θ < π/2. Suppose f (jω) = f (−jω). Show that Z ∞ f (rejθ ) = f (jω)Kθ (ω/r) d(ln ω) −∞ where Kθ (ω/r) = 1 2(ω/r)[1 + (ω/r)2 ] cos θ π [1 − (ω/r)2 ]2 + 4(ω/r)2 cos2 θ Chapter 17 Gap Metric and ν-Gap Metric In the previous chapters, we have seen that all of robust control design techniques assume that we have some description of the model uncertainties (i.e., we have a measure of the distance from the nominal plant to the set of uncertainty systems). This measure is usually chosen to be a metric or a norm. However, the operator norm can be a poor measure of the distance between systems in respect to feedback control system design. For example, consider 1 1 . P1 (s) = , P2 (s) = s s + 0.1 The closed-loop complementary sensitivity functions corresponding to P1 and P2 with unity feedback are relatively close and their difference is P1 (I + P1 )−1 − P2 (I + P2 )−1 ∞ = 0.0909, but the difference between P1 and P2 is kP1 − P2 k∞ = ∞. This shows that the closed-loop behavior of two systems can be very close even though the norm of the difference between the two open-loop systems can be arbitrarily large. To deal with such problems, the gap metric and the ν-gap metric were introduced into the control literature by Zames and El-Sakkary [1980] (see also El-Sakkary [1985] and Vinnicombe [1993]) as being appropriate for the study of uncertainty in feedback systems. An alternative metric, the graph metric, was also introduced by Vidyasagar [1984] in terms of normalized coprime factorizations. All of these metrics are equivalent, and thus induce the same topology. This topology is the weakest in which feedback stability is a robust property. The metrics define notions of distance in the space of (possible) unstable systems that do not assume that the plants have the same number of poles in the right-half plane. We shall briefly introduce the gap metric in Section 17.1 and study some of its applications in robust control. Our focus in this chapter is Sections 17.2–17.4, which 349 350 GAP METRIC AND ν-GAP METRIC study in some detail the ν-gap metric. In particular, we introduce the ν-gap metric in Section 17.2 and explore its frequency domain interpretation and applications in Section 17.3 and Section 17.4. Finally, we consider controller order reduction in the gap or ν-gap metric framework in Section 17.5. 17.1 Gap Metric In this section we briefly introduce the gap metric and discuss some of its applications in controller design. Let P (s) be a p × m rational transfer matrix and let P have the following normalized right and left stable coprime factorizations: P = N M −1 = M̃ −1 Ñ . That is, M ∼ M + N ∼ N = I, M̃ M̃ ∼ + Ñ Ñ ∼ = I. The graph of the operator P is the subspace of H2 consisting of all pairs (u, y) such that y = P u. This is given by " # M H2 N and is a closed subspace of H2 . The gap between two systems P1 and P2 is defined by δg (P1 , P2 ) = Π" M1 N1 # H2 − Π" # M2 N2 H2 where ΠK denotes the orthogonal projection onto K and P1 = N1 M1−1 and P2 = N2 M2−1 are normalized right coprime factorizations. It is shown by Georgiou [1988] that the gap metric can be computed as follows: Theorem 17.1 Let P1 = N1 M1−1 and P2 = N2 M2−1 be normalized right coprime factorizations. Then n o δg (P1 , P2 ) = max ~δ(P1 , P2 ), ~δ(P2 , P1 ) where ~δg (P1 , P2 ) is the directed gap and can be computed by ~δg (P1 , P2 ) = inf Q∈H∞ " M1 N1 # − " M2 N2 # . Q ∞ 17.1. Gap Metric 351 The following procedures can be used in computing the directed gap ~δg (P1 , P2 ). Computing ~ δg (P1 , P2 ): Let " P1 = A1 B1 C1 D1 # , P2 = " A2 B2 C2 D2 # . 1. Let Pi = Ni Mi−1 , i = 1, 2 be normalized right coprime factorizations. Then " where Mi Ni #   = −1/2 Ai + Bi Fi Bi Ri Fi Ci + Di Fi −1/2 Ri −1/2 Di Ri Xi = Ric "  Ri = I + Di∗ Di R̃i = I + Di Di∗ Fi = −Ri−1 (Bi∗ Xi + Di∗ Ci )  , Ai − Bi Ri−1 Di∗ Ci −Ci∗ R̃i−1 Ci −Bi Ri−1 Bi∗ −(Ai − Bi Ri−1 Di∗ Ci )∗ # . 2. Define a generalized system  "   G(s) =  A1 + B1 F1  0    = F1   C1 + D1 F1 0 M1 N1 −I 0 A2 + B2 F2 F2 C2 + D2 F2 0 # " #  M2 N2   0 −1/2 B1 R1 0 −1/2 R1 −1/2 D1 R1 −I 3. Apply standard H∞ algorithm to # # " " M2 M1 ~δg (P1 , P2 ) = inf Q − Q∈H∞ N2 N1 ∞  0 −1/2  B2 R2   −1/2 . R2  −1/2  D2 R2  0 = inf kFℓ (G, Q)k∞ . Q∈H∞ Using the above procedure, it is easy to show that   1 1 = 0.0995, , δg s s + 0.1 which confirms that the two systems given at the beginning of this chapter are indeed close in the gap metric. This example shows an important feature about the gap metric (similarly, the ν-gap metric defined in the next section): The distance between two 352 GAP METRIC AND ν-GAP METRIC plants, as measured by the gap metric δg (or the ν-gap metric δν ), has very little to do with any difference between their open-loop behavior (indeed, there is no reason why it should). This point will be further illustrated by an example in the next section. A lower bound for the gap metric can also be obtained easily without actually solving the corresponding H∞ optimization. Let " # M2∼ N2∼ Φ= . −Ñ2 M̃2 Then Φ∼ Φ = ΦΦ∼ = I and ~δg (P1 , P2 ) = = inf " inf " Q∈H∞ Q∈H∞ M1 N1 # M2∼ M1 + N2∼ N1 − Q −Ñ2 M1 + M̃2 N1 # M2∼ −Ñ2 N2∼ M̃2 # (" − " ∞ M2 N2 # ) Q ∞ ≥ kΨ(P1 , P2 )k∞ where Ψ(P1 , P2 ) := −Ñ2 M1 + M̃2 N1 = h M̃2 Ñ2 i " 0 −I I 0 #" M1 N1 # . (17.1) It will be seen in the next section that kΨ(P1 , P2 )k∞ is related to the ν-gap metric. The above lower bound may actually be achieved. Consider, for example, P1 = k1 , s+1 P2 = k2 . s+1 Then it is easy to verify that Pi = Ni /Mi , i = 1, 2, with Ni = k pi , s + 1 + ki2 Mi = s+1 p , s + 1 + ki2 are normalized coprime factorizations and it can be further shown, as in Georgiou and Smith [1990], that  |k1 − k2 |   , if |k1 k2 | > 1;   |k  1 | + |k2 | δg (P1 , P2 ) = kΨ(P1 , P2 )k∞ =   |k1 − k2 |   , if |k1 k2 | ≤ 1.  p (1 + k12 )(1 + k22 ) An immediate consequence of Theorem 17.1 is the connection between the uncertainties in the gap metric and the uncertainties characterized by the normalized coprime factors. The following corollary states that a ball of uncertainty in the directed gap is equivalent to a ball of uncertainty in the normalized coprime factors. 17.1. Gap Metric 353 Corollary 17.2 Let P have a normalized coprime factorization P = N M −1 . Then for all 0 < b ≤ 1, n o P1 : ~δg (P, P1 ) < b = ( −1 P1 : P1 = (N + ∆N )(M + ∆M ) , ∆N , ∆M ∈ H∞ , " ∆N ∆M # ) <b . ∞ Proof. Suppose ~δg (P, P1 ) < b and let P1 = N1 M1−1 be a normalized right coprime factorization. Then there exists a Q ∈ H∞ such that " Define " " ∆M ∆M ∆N # M N # := " − " # Q Q− " M1 N1 # M1 N1 < b. ∞ M N # ∈ H∞ . # < b and P1 = (N1 Q)(M1 Q)−1 = (N + ∆N )(M + ∆M )−1 . ∆N ∞ To show the converse, note that P1 = (N + ∆N )(M + ∆M )−1 and there exists a n on o−1 Q̃−1 ∈ H∞ such that P1 = (N + ∆N )Q̃ (M + ∆M )Q̃ is a normalized right coprime factorization. Hence by definition, ~δg (P, P1 ) can be computed as Then ~δg (P, P1 ) = inf Q " M N # − " M + ∆M N + ∆N # Q̃Q ∞ ≤ " M N # − " M + ∆M N + ∆N where the first inequality follows by taking Q = Q̃−1 ∈ H∞ . # <b ∞ ✷ The following is a list of useful properties of the gap metric shown by Georgiou and Smith [1990]. • If δg (P1 , P2 ) < 1, then δg (P1 , P2 ) = ~δg (P1 , P2 ) = ~δg (P2 , P1 ). • If b ≤ λ(P ) := inf σ Res>0 " M (s) N (s) #! , then n o P1 : ~δ(P, P1 ) < b = {P1 : δ(P, P1 ) < b} . 354 GAP METRIC AND ν-GAP METRIC Recall that bobt (P ) := = and bP,K := " I K # ( " inf K stabilizing I K p 1 − λmax (Y Q) = −1 (I + P K) h I P i # −1 (I + P K) r h 1− " −1 = ∞ I P Ñ # h I i M̃ i P 2 ∞ )−1 H −1 (I + KP ) h I K i −1 . ∞ The following results were shown by Qiu and Davison [1992a]. Theorem 17.3 Suppose the feedback system with the pair (P0 , K0 ) is stable. Let P := {P : δg (P, P0 ) < r1 } and K := {K : δg (K, K0 ) < r2 }. Then (a) The feedback system with the pair (P, K) is also stable for all P ∈ P and K ∈ K if and only if arcsin bP0 ,K0 ≥ arcsin r1 + arcsin r2 . (b) The worst possible performance resulting from these sets of plants and controllers is given by inf P ∈P, K∈K arcsin bP,K = arcsin bP0 ,K0 − arcsin r1 − arcsin r2 . The sufficiency part of the theorem follows from Theorem 17.8 in the next section. Note that the theorem is still true if one of the uncertainty balls is taken as closed ball. In particular, one can take either r1 = 0 or r2 = 0. Example 17.1 Consider P1 = s−1 , s+1 P2 = 2s − 1 . s+1 Then P1 = N1 /M1 and P2 = N2 /M2 with 1 s−1 , N1 = √ 2s+1 1 M1 = √ , 2 N2 = √ 2s − 1 √ , 5s + 2 s+1 √ M2 = √ 5s + 2 are normalized coprime factorizations. It is easy to show that |ω| 1 δg (P1 , P2 ) = 1/3 > kΨ(P1 , P2 )k∞ = sup √ =√ , 2 ω 10ω + 4 10 17.1. Gap Metric 355 which implies that any controller K that stabilizes P1 and achieves only bP1 ,K > 1/3 will actually stabilize P2 by Theorem 17.3. The following Matlab command can be used to compute the gap: ≫ δg (P1 , P2 ) = gap(P1 , P2 , tol) √ Next, note that bobt (P1 ) = 1/ 2 and the optimal controller√achieving bobt (P1 ) is Kobt = 0. There must be a plant P with δν (P1 , P ) = bobt (P1 ) = 1/ 2 that can not be stabilized by√ Kobt = 0; that is, there must be an unstable plant P such that δν (P1 , P ) = bobt (P1 ) = 1/ 2. A such P can be found using Corollary 17.2: {P : δg (P1 , P ) ≤ bobt (P1 )} # ) " ( ∆N N1 + ∆N ≤ bobt (P1 ) . , ∆N , ∆M ∈ H∞ , = P : P = M1 + ∆M ∆M ∞ " # ∆N that is, there must be ∆N , ∆M ∈ H∞ , = bobt (P1 ) such that ∆M ∞ N1 + ∆N M1 + ∆M P = is unstable. Let ∆N = 0, Then P = 1 s−1 . ∆M = √ 2s+1 s−1 N1 + ∆N = , M1 + ∆M 2s √ δν (P1 , P ) = bobt (P1 ) = 1/ 2. Example 17.2 We shall now consider the following question: Given an uncertain plant P (s) = k , s−1 k ∈ [k1 , k2 ], (a) Find the best nominal design model P0 = inf k0 ∈[k1 ,k2 ] k0 in the sense s−1 sup δg (P, P0 ). k∈[k1 ,k2 ] (b) Let k1 be fixed and k2 be variable. Find the k0 so that the largest family of the plant P can be guaranteed to be stabilized a priori by any controller satisfying bP0 ,K = bobt (P0 ). 356 GAP METRIC AND ν-GAP METRIC For simplicity, suppose k1 ≥ 1. It can be shown that δg (P, P0 ) = optimal k0 for question (a) satisfies that is, k0 = |k0 −k| k0 +k . Then the k0 − k1 k2 − k0 = ; k0 + k1 k2 + k0 √ k1 k2 and inf k0 ∈[k1 ,k2 ] √ √ k2 − k1 √ . δg (P, P0 ) = √ k2 + k1 k∈[k1 ,k2 ] sup To answer question (b), we note that by Theorem 17.3, a family of plants satisfying δg (P, P0 ) ≤ r with P0 = k0 /(s + 1) is stabilizable a priori by any controller satisfying bP0 ,K = bobt (P0 ) if, and only if, r < bP0 ,K . Since P0 = N0 /M0 with N0 = s+ k0 p , 1 + k02 M0 = s+ s−1 p 1 + k02 is a normalized coprime factorization, it is easy to show that q p " # k02 + (1 − 1 + k02 )2 N0 p = M0 2 1 + k02 H and v u u1 bobt (P0 ) = t 2 Hence we need to find a k0 such that bobt (P0 ) ≥ max that is, v u u1 t 2 1 1+ p 1 + k02 !  1 ! 1+ p . 1 + k02 k0 − k1 k2 − k0 , k0 + k1 k2 + k0 ≥ max   ; k0 − k1 k2 − k0 , k0 + k1 k2 + k0  for a largest possible k2 . The optimal k0 is given by the solution of the equation: v ! u u1 k0 − k1 1 t = 1+ p 2 2 k 0 + k1 1 + k0 and the largest k2 = k02 /k1 . For example, if k1 = 1, then k0 = 7.147 and k2 = 51.0793. In general, given a family of plant P , it is not easy to see how to choose a best nominal model P0 such that (a) or (b) is true. This is still a very important open question. 17.2. ν-Gap Metric 17.2 357 ν-Gap Metric The shortfall of the gap metric is that it is not easily related to the frequency response of the system. On the other hand, the ν-gap metric to be introduced in this section has a clear frequency domain interpolation and can, in general, be computed from frequency response. The presentation given in this section, Sections 17.3, 17.4, and 17.5 are based on Vinnicombe [1993a, 1993b], to which readers are referred for further detailed discussions. To define the new metric, we shall first review a basic concept from the complex analysis. Definition 17.1 Let g(s) be a scalar transfer function and let Γ denote a Nyquist contour indented around the right of any imaginary axis poles of g(s), as shown in Figure 17.1. Then the winding number of g(s) with respect to this contour, denoted by wno(g), is the number of counterclockwise encirclements around the origin by g(s) evaluated on the Nyquist contour Γ. (A clockwise encirclement counts as a negative encirclement.) Γ 0 Figure 17.1: The Nyquist contour The following argument principle is standard and can be found from any complex analysis book. Lemma 17.4 (The Argument Principle) Let Γ be a closed contour in the complex plane. Let f (s) be a function analytic along the contour; that is, f (s) has no poles on Γ. Assume f (s) has Z zeros and P poles inside Γ. Then f (s) evaluated along the contour Γ once in an anticlockwise direction will make Z − P anticlockwise encirclements of the origin. 358 GAP METRIC AND ν-GAP METRIC Let G(s) be a matrix (or scalar) transfer matrix. We shall denote η(G) and η0 (G), respectively, the number of open right-half plane and imaginary axis poles of G(s). The winding number has the following properties: Lemma 17.5 Let g and h be biproper rational scalar transfer functions and let F be a square transfer matrix. Then (a) wno(gh) = wno(g)+wno(h); (b) wno(g) = η(g −1 ) − η(g); (c) wno(g ∼ ) = −wno(g) − η0 (g −1 ) + η0 (g); (d) wno(1 + g) = 0 if g ∈ RL∞ and kgk∞ < 1; (e) wno det(I + F ) = 0 if F ∈ RL∞ and kF k∞ < 1. Proof. Part (a) is obvious by the argument principle. To show part (b), note that by the argument principle wno(g) equals the excess of the number of open right-half plane zeros of g over the number of open right-half plane poles of g; that is, wno(g) = η(g −1 )− η(g), since the number of right-half plane zeros of g is the number of right-half plane −1 poles of . Next, suppose the order of g in part (c) is n. Then η(g ∼ ) = n−η(g)−η 0 (g)  g∼ −1   and η (g ) = n − η(g −1 ) − η0 (g −1 ), which gives wno(g ∼ ) = η (g ∼ )−1 − η(g ∼ ) = η(g) − η(g −1 ) − η0 (g −1 ) + η0 (g) = −wno(g) − η0 (g −1 ) + η0 (g). Part (d) follows from the fact that 1 + Reg(jω) Qm > 0, ∀ω since kgk∞ < 1. Finally, part (e) follows from part (d) and det(I + F ) = i=1 (1 + λi (F )) with |λi (F )| < 1. ✷ Example 17.3 Let g1 = 1.2(s + 3) , s−5 g2 = s−1 , s−2 g3 = 2(s − 1)(s − 2) , (s + 3)(s + 4) g4 = (s − 1)(s + 3) . (s − 2)(s − 4) Figure 17.2 shows the functions, g1 , g2 , g3 , and g4 , evaluated on the Nyquist contour Γ. Clearly, we have wno(g1 ) = −1, wno(g2 ) = 0, wno(g3 ) = 2, wno(g4 ) = −1 and they are consistent with the results computed from using Lemma 17.5. The ν-gap metric introduced in Vinnicombe [1993a, 1993b] is defined as follows: 17.2. ν-Gap Metric 359 2 1.5 1 g1 g2 g3 g4 imaginary 0.5 0 −0.5 −1 −1.5 −2 −1 −0.5 0 0.5 real 1 1.5 Figure 17.2: g1 , g2 , g3 , and g4 evaluated on Γ Definition 17.2 The ν-gap metric is defined as   kΨ(P1 , P2 )k∞ , if det Θ(jω) 6= 0 ∀ω     and wno det Θ(s) = 0, δν (P1 , P2 ) =      1, otherwise where Θ(s) := N2∼ N1 + M2∼ M1 and Ψ(P1 , P2 ) := −Ñ2 M1 + M̃2 N1 . Note that it can be shown as in Vinnicombe [1993a] that δν (P1 , P2 ) = δν (P2 , P1 ) = δν (P1T , P2T ) and δν is indeed a metric (a proof of this fact is quite complex). 2 360 GAP METRIC AND ν-GAP METRIC Computing δν (P1 , P2 ): Let " P1 = A1 B1 C1 D1 # , P2 = " A2 B2 C2 D2 # . 1. Let Pi = Ni Mi−1 , i = 1, 2 be normalized right coprime factorizations. Then " where Mi Ni #   = −1/2 Ai + Bi Fi Bi Ri Fi Ci + Di Fi −1/2 Ri −1/2 Di Ri Xi = Ric "  Ri = I + Di∗ Di R̃i = I + Di Di∗  , Ai − Bi Ri−1 Di∗ Ci Fi = −Ri−1 (Bi∗ Xi + Di∗ Ci ) −Bi Ri−1 Bi∗ −Ci∗ R̃i−1 Ci −(Ai − Bi Ri−1 Di∗ Ci )∗ # . 2. Compute the zeros of Θ(s) := N2∼ N1 + M2∼ M1 = " M2 N2 #∼ " M1 N1 # . Let n0 = number of imaginary axis zeros of Θ, n+ = number of open right-half plane zeros of Θ, and n = number of open right-half plane poles of Θ. Then wno det(N2∼ N1 + M2∼ M1 ) = n+ − n. 3. If either n0 6= 0 or n+ 6= n, δν (P1 , P2 ) = 1. Otherwise, δν (P1 , P2 ) = kΨ(P1 , P2 )k∞ with Ψ(P1 , P2 ) = −Ñ2 M1 + M̃2 N1 : # # " " Li Bi + Li Di Ai + Li Ci M̃i = −1/2 −1/2 −1/2 Ñi Di Ci R̃i R̃i R̃i Li = −(Bi Di∗ + Yi Ci∗ )R̃i−1 where Yi = Ric " (Ai − Bi Di∗ R̃i−1 Ci )∗ −Bi Ri−1 Bi∗ −Ci∗ R̃i−1 Ci −(Ai − Bi Di∗ R̃i−1 Ci ) # . The Matlab command nugap can be used to carry out the preceding computation: ≫ δν (P1 , P2 ) = nugap(P1 , P2 , tol) where tol is the computational tolerance. 17.2. ν-Gap Metric 361 Example 17.4 Consider, for example, P1 = 1 and P2 = 1 M 1 = N1 = √ , 2 Hence M2 = 1 1 1−s =√ , Θ(s) = √ 1 − s 2 2 and δν (P1 , P2 ) = √1 . 2 s , s+1 1 . Then s N2 = 1 . s+1 1 s−1 Ψ(P1 , P2 ) = √ , 2s+1 (Note that Θ has no poles or zeros!) The ν-gap metric can also be computed directly from the system transfer matrices without first finding the normalized coprime factorizations. Theorem 17.6 The ν-gap metric can be defined as   kΨ(P1 , P2 )k∞ , if det(I + P2∼ P1 ) 6= 0 ∀ω and     wno det(I + P2∼ P1 ) + η(P1 ) − η(P2 ) − η0 (P2 ) = 0, δν (P1 , P2 ) =      1, otherwise where Ψ(P1 , P2 ) can be written as Ψ(P1 , P2 ) = (I + P2 P2∼ )−1/2 (P1 − P2 )(I + P1∼ P1 )−1/2 . Proof. Since the number of unstable zeros of M1 (M2 ) is equal to the number of unstable poles of P1 (P2 ), and N2∼ N1 + M2∼ M1 = M2∼ (I + P2∼ P1 )M1 , we have wno det(N2∼ N1 + M2∼ M1 ) = wno det {M2∼ (I + P2∼ P1 )M1 } = wno det M2∼ + wno det(I + P2∼ P1 ) + wno det M1 . Note that wno det M1 = η(P1 ), wno det M2∼ = −wno det M2 − η0 (M2−1 ) = −η(P2 ) − η0 (P2 ), and wno det(N2∼ N1 + M2∼ M1 ) = −η(P2 ) − η0 (P2 ) + wno det(I + P2∼ P1 ) + η(P1 ). Furthermore, det(N2∼ N1 + M2∼ M1 ) 6= 0, ∀ω ⇐⇒ det(I + P2∼ P1 ) 6= 0, ∀ω. 362 GAP METRIC AND ν-GAP METRIC The theorem follows by noting that Ψ(P1 , P2 ) = (I + P2 P2∼ )−1/2 (P1 − P2 )(I + P1∼ P1 )−1/2 since Ψ(P1 , P2 ) = −Ñ2 M1 + M̃2 N1 = M̃2 (P1 − P2 )M1 and M̃2∼ M̃2 = (I + P2 P2∼ )−1 , M1 M1∼ = (I + P1∼ P1 )−1 . ✷ This alternative formula is useful when doing the hand calculation or when computing from the frequency response of the plants since it does not need to compute the normalized coprime factorizations. Example 17.5 Consider two plants P1 = 1 and P2 = 1/s. Then wno det(1 + P2∼ P1 ) = wno[(s − 1)/s] = 1, as shown in Figure 17.3(a), and wno det(1 + P2∼ P1 ) + η(P1 ) − η(P2 ) − η0 (P2 ) = 0. On the other hand, wno det(1 + P1∼ P2 ) + η(P2 ) − η(P1 ) = wno (s + 1)/s = 0, as shown in Figure 17.3(b). (s+1)/s 10 10 5 5 (s−1)/s 0 0 −5 −5 −10 −10 −10 −5 0 (a) Figure 17.3: 0 5 10 (b) s−1 s+1 and evaluated on Γ s s Similar to the gap metric, it is shown by Vinnicombe [1993a, 1993b] that the ν-gap metric can also be characterized as an optimization problem (however, we shall not use it for computation). 17.2. ν-Gap Metric 363 Theorem 17.7 Let P1 = N1 M1−1 and P2 = N2 M2−1 be normalized right coprime factorizations. Then " # " # M1 M2 − . Q δν (P1 , P2 ) = inf N1 N2 Q, Q−1 ∈ L∞ ∞ wno det(Q) = 0 Moreover, δg (P1 , P2 )bobt (P1 ) ≤ δν (P1 , P2 ) ≤ δg (P1 , P2 ). It is now easy to see that ⊃ {P : δν (P0 , P ) < r} # " ∆N −1 ∈ H∞ , P = (N0 + ∆N )(M0 + ∆M ) : ∆M " ( Define 1 bP,K (ω) := σ " I K(jω) # −1 (I + P (jω)K(jω)) h ∆N ∆M I # <r ∞ P (jω) i ) . ! and ψ(P1 (jω), P2 (jω)) = σ (Ψ(P1 (jω), P2 (jω))) . The following theorem states that robust stability can be checked using the frequencyby-frequency test. Theorem 17.8 Suppose (P0 , K) is stable and δν (P0 , P1 ) < 1. Then (P1 , K) is stable if bP0 ,K (ω) > ψ(P0 (jω), P1 (jω)), ∀ω. Moreover, arcsin bP1 ,K (ω) ≥ arcsin bP0 ,K (ω) − arcsin ψ(P0 (jω), P1 (jω)), ∀ω and arcsin bP1 ,K ≥ arcsin bP0 ,K − arcsin δν (P0 , P1 ). Proof. Let P1 = M̃1−1 Ñ1 , P0 = N0 M0−1 = M̃0−1 Ñ0 and K = U V −1 be normalized coprime factorizations, respectively. Then ! " # i h   V 1 −1 =σ = σ (M̃1 V + Ñ1 U )−1 . (M̃1 V + Ñ1 U ) M̃1 Ñ1 bP1 ,K (ω) U That is, bP1 ,K (ω) = σ(M̃1 V + Ñ1 U ) = σ h M̃1 Ñ1 i " V U #! . 364 GAP METRIC AND ν-GAP METRIC Similarly, h bP0 ,K (ω) = σ(M̃0 V + Ñ0 U ) = σ Ñ0 M̃0 Note that h ψ(P0 (jω), P1 (jω)) = σ " N0 M̃0∼ −M0 Ñ0∼ To simplify the derivation, define # " h N0 , G̃0 = M̃0 G0 = −M0 #∼ " Ñ0 i M̃1 Ñ1 i N0 M̃0∼ −M0 Ñ0∼ , G̃1 = h " # i " N0 −M0 V U #! . #! = I. M̃1 Ñ1 i , F = " V U # . Then ψ(P0 , P1 ) = σ(G̃1 G0 ), and That is, h G0 G̃∼ 0 i∼ h G0 bP0 ,K (ω) = σ(G̃0 F ), G̃∼ 0 i = I =⇒ h G0 bP1 ,K (ω) = σ(G̃1 F ) G̃∼ 0 ih G0 G̃∼ 0 i∼ = I. ∼ G0 G∼ 0 + G̃0 G̃0 = I. Note that ∼ ∼ ∼ ∼ ∼ ∼ ∼ I = G̃1 G̃∼ 1 = G̃1 (G0 G0 + G̃0 G̃0 )G̃1 = (G̃1 G0 )(G̃1 G0 ) + (G̃1 G̃0 )(G̃1 G̃0 ) . Hence 2 σ2 (G̃1 G̃∼ 0 ) = 1 − σ (G̃1 G0 ). Similarly, ∼ ∼ ∼ ∼ ∼ I = F ∼ F = F ∼ (G0 G∼ 0 + G̃0 G̃0 )F = (G0 F ) (G0 F ) + (G̃0 F ) (G̃0 F ) 2 =⇒ σ2 (G∼ 0 F ) = 1 − σ (G̃0 F ). By the assumption, ψ(P0 , P1 ) < bP0 ,K (ω); that is, σ(G̃1 G0 ) < σ(G̃0 F ), and σ(G∼ 0 F) = Hence ∀ω q q 1 − σ2 (G̃0 F ) < 1 − σ 2 (G̃1 G0 ) = σ(G̃1 G̃∼ 0 ). ∼ σ(G̃1 G0 )σ(G∼ 0 F ) < σ(G̃1 G̃0 )σ(G̃0 F ); 17.2. ν-Gap Metric 365 that is, ∼ σ(G̃1 G0 G∼ 0 F ) < σ(G̃1 G̃0 G̃0 F ), −1 (G̃1 G0 G∼ =⇒ (G̃1 G̃∼ 0 G0 F ) 0 F) Now ∀ω ∞ < 1. ∼ ∼ ∼ G̃1 F = G̃1 (G̃∼ 0 G̃0 + G0 G0 )F = (G̃1 G̃0 G̃0 F ) + (G̃1 G0 G0 F )   ∼ −1 (G̃1 G0 G∼ = (G̃1 G̃∼ 0 F) . 0 G̃0 F ) I + (G̃1 G̃0 G̃0 F ) By Lemma 17.5, ∼ wno det(G̃1 F ) = wno det(G̃1 G̃∼ 0 G̃0 F ) = wno det(G̃1 G̃0 ) + wno det(G̃0 F ). Since (P0 , K) is stable =⇒ (G̃0 F )−1 ∈ H∞ =⇒ η((G̃0 F )−1 ) = 0 =⇒ wno det(G̃0 F ) := η((G̃0 F )−1 ) − η(G̃0 F ) = 0. Next, note that P0T = (Ñ0T )(M̃0T )−1 , P1T = (Ñ1T )(M̃1T )−1 and δν (P0T , P1T ) = δν (P0 , P1 ) < 1; then, by definition of δν (P0T , P1T ), ∼ T wno det((Ñ0T )∼ (Ñ1T ) + (M̃0T )∼ (M̃1T )) = wno det(G̃1 G̃∼ 0 ) = wno det(G̃1 G̃0 ) = 0. Hence wno det(G̃1 F ) = 0, but wno det(G̃1 F ) := η((G̃1 F )−1 ) − η(G̃1 F ) = η((G̃1 F )−1 ) since η(G̃1 F ) = 0, so η((G̃1 F )−1 ) = 0; that is, (P1 , K) is stable. Finally, note that ∼ ∼ ∼ G̃1 F = G̃1 (G̃∼ 0 G̃0 + G0 G0 )F = (G̃1 G̃0 )(G̃0 F ) + (G̃1 G0 )(G0 F ) and ∼ σ(G̃1 F ) ≥ σ(G̃1 G̃∼ 0 )σ(G̃0 F ) − σ(G̃1 G0 )σ(G0 F ) q q = 1 − σ 2 (G̃1 G0 )σ(G̃0 F ) − σ(G̃1 G0 ) 1 − σ2 (G̃0 F ) = sin(arcsin σ(G̃0 F ) − arcsin σ(G̃1 G0 )) = sin(arcsin bP0 ,K (ω) − arcsin ψ(P0 (jω), P1 (jω))) and, consequently, arcsin bP1 ,K (ω) ≥ arcsin bP0 ,K (ω) − arcsin ψ(P0 (jω), P1 (jω)) and inf arcsin bP1 ,K (ω) ≥ inf arcsin bP0 ,K (ω) − sup arcsin ψ(P0 (jω), P1 (jω)). ω ω ω That is, arcsin bP1 ,K ≥ arcsin bP0 ,K − arcsin δν (P0 , P1 ). ✷ 366 GAP METRIC AND ν-GAP METRIC The significance of the preceding theorem can be illustrated using Figure 17.4. It is clear from the figure that δν (P0 , P1 ) > bP0 ,K . Thus a frequency-independent stability test cannot conclude that a stabilizing controller K for P0 will stabilize P1 . However, the frequency-dependent test in the preceding theorem shows that K stabilizes both P0 and P1 since bP0 ,K (ω) > ψ(P0 (jω), P1 (jω)) for all ω. Furthermore, bP1 ,K ≥ inf sin (arcsin bP0 ,K (ω) − arcsin ψ(P0 , P1 )) > 0. ω b P , K (ω) 0 b P ,K 0 δ v(P0 , P1 ) jω)) ψ(P0 (jω), P( 1 ω Figure 17.4: K stabilizes both P0 and P1 since bP0 ,K (ω) > ψ(P0 , P1 ) for all ω The following theorem is one of the main results on the ν-gap metric. Theorem 17.9 Let P0 be a nominal plant and β ≤ α < bobt (P0 ). (i) For a given controller K, arcsin bP,K > arcsin α − arcsin β for all P satisfying δν (P0 , P ) ≤ β if and only if bP0 ,K > α. (ii) For a given plant P , arcsin bP,K > arcsin α − arcsin β for all K satisfying bP0 ,K > α if and only if δν (P0 , P ) ≤ β. Proof. The sufficiency follows essentially from Theorem 17.8. The necessity proof is harder, see Vinnicombe [1993a, 1993b] for details. ✷ The preceding theorem shows that any plant at a distance less than β from the nominal will be stabilized by any controller stabilizing the nominal with a stability 17.2. ν-Gap Metric 367 margin of β. Furthermore, any plant at a distance greater than β from the nominal will be destabilized by some controller that stabilizes the nominal with a stability margin of at least β. Similarly, one can consider the system robust performance with simultaneous perturbations on the plant and controller. Theorem 17.10 Suppose the feedback system with the pair (P0 , K0 ) is stable. Then arcsin bP,K ≥ arcsin bP0 ,K0 − arcsin δν (P0 , P ) − arcsin δν (K0 , K) for any P and K. Proof. Use the fact that bP,K = bK,P and apply Theorem 17.8 to get arcsin bP,K ≥ arcsin bP0 ,K − arcsin δν (P0 , P ). Dually, we have arcsin bP0 ,K ≥ arcsin bP0 ,K0 − arcsin δν (K0 , K). Hence the result follows. ✷ Example 17.6 Consider again the following example, studied in Vinnicombe [1993b], with 2s − 1 s−1 , P2 = P1 = s+1 s+1 and note that −2s − 1 s − 1 3s + 2 1 + P2∼ P1 = 1 + = . −s + 1 s + 1 s+1 Then 1 + P2∼ (jω)P1 (jω) 6= 0, ∀ω, wno det(I + P2∼ P1 ) + η(P1 ) − η(P2 ) = 0 and 1 |ω| |P1 − P2 | p =√ . = sup √ δν (P1 , P2 ) = kΨ(P1 , P2 )k∞ = sup p 10ω 2 + 4 10 ω ω 1 + |P1 |2 1 + |P2 |2 √ This implies that any controller K that stabilizes P1 and achieves only bP1 ,K > 1/ 10 will actually stabilize P2 . This result is clearly less conservative than that√of using the gap metric. Furthermore, there exists a controller such that bP1 ,K = 1/ 10 that destabilizes P2 . Such a controller is K = −1/2, which results in a closed-loop system with P2 illposed. 368 GAP METRIC AND ν-GAP METRIC Example 17.7 Consider the following example taken from Vinnicombe [1993b]: P1 = 100 , 2s + 1 P2 = 100 , 2s − 1 P3 = 100 . (s + 1)2 P1 and P2 have very different open-loop characteristics—one is stable, the other unstable. However, it is easy to show that δν (P1 , P2 ) = δg (P1 , P2 ) = 0.02, δν (P1 , P3 ) = δg (P1 , P3 ) = 0.8988, δν (P2 , P3 ) = δg (P2 , P3 ) = 0.8941, which show that P1 and P2 are very close while P1 and P3 (or P2 and P3 ) are quite far away. It is not surprising that any reasonable controller for P1 will do well for P2 but not necessarily for P3 . The closed-loop step responses under unity feedback, K1 = 1, are shown in Figure 17.5. 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 P1 P2 0.2 P3 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 17.5: Closed-loop step responses with K1 = 1 The corresponding stability margins for the closed-loop systems with P1 and P2 are bP1 ,K1 = 0.7071, and bP2 ,K1 = 0.7, respectively, which are very close to their maximally possible margins, bobt (P1 ) = 0.7106, and bobt (P2 ) = 0.7036 17.2. ν-Gap Metric 369 (in fact, the optimal controllers for P1 and P2 are K = 0.99 and K = 1.01, respectively). While the stability margin for the closed-loop system with P3 is bP3 ,K1 = 0.0995, which is far away from its optimal value, bobt (P3 ) = 0.4307, and results in poor performance of the closed loop. In fact, it is not hard to find a controller that will perform well for both P1 and P2 but will destabilize P3 . Of course, this does not necessarily mean that all controllers performing reasonably well with P1 and P2 will do badly with P3 , merely that some do — the unit feedback being an example. It may be harder to find a controller that will perform reasonably well with all three plants; the maximally stabilizing controller of P3 , K3 = 2.0954s + 10.8184 , s + 23.2649 is a such controller, which gives bP1 ,K3 = 0.4307, and bP3 ,K3 = 0.4307. bP2 ,K3 = 0.4126, The step responses under this control law are shown in Figure 17.6. 1.2 1 0.8 0.6 P1 P2 0.4 P3 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 17.6: Closed-loop step responses with K3 = 0.8 0.9 1 2.0954s + 10.8184 s + 23.2649 370 17.3 GAP METRIC AND ν-GAP METRIC Geometric Interpretation of ν-Gap Metric The most salient feature of the ν-gap metric is that it can be computed pointwise in frequency domain: δν (P1 , P2 ) = sup ψ(P1 (jω), P2 (jω)) ω provided the winding number condition is satisfied. (For a more extensive coverage of material presented in this section and the next two sections, readers are encouraged to consult the original references by Vinnicombe [1992a, 1992b, 1993a, 1993b].) In particular, for a single-input single-output system, |P1 (jω) − P2 (jω)| p . ψ(P1 (jω), P2 (jω)) = p 1 + |P1 (jω)|2 1 + |P2 (jω)|2 (17.2) This function has the interpretation of being the chordal distance between P1 (jω) and P2 (jω). To illustrate this, consider the Riemann sphere, which is a unit sphere tangent at its “south pole” to the complex plant at its origin shown in Figure 17.7. Figure 17.7: Projection onto the Riemann sphere A point s1 (e.g., s1 = 1−j) in the complex plane is stereographically projected on the Riemann sphere by connecting the “north pole” to s1 and determining the intersection of this straight line with the Riemann sphere, resulting in the projection, q1 , of s1 . The 17.3. Geometric Interpretation of ν-Gap Metric 371 coordinates of q1 are x1 = Res1 , 1 + |s1 |2 y1 = ℑs1 , 1 + |s1 |2 z1 = |s1 |2 . 1 + |s1 |2 Thus, the north pole represents the point at infinity and the unit circle is projected onto the “equator.” The chordal distance between two points, s1 and s2 , is the Euclidean distance between their stereographical projections, q1 and q2 : d(s1 , s2 ) = p |s1 − s2 | p . (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 = p 1 + |s1 |2 1 + |s2 |2 1 0.8 0.6 0.4 0.2 0 −0.5 0 0.5 0.5 0 1 1.5 real axis −0.5 imaginary axis Figure 17.8: Projection of a disk on the Nyquist diagram onto the Riemann sphere Now consider a circle of chordal radius r centered at P0 (jω0 ) on the Riemann sphere for some frequency ω0 ; that is, |P (jω0 ) − P0 (jω0 )| p p = r. 1 + |P (jω0 )|2 1 + |P0 (jω0 )|2 Let P (jω0 ) = R + jI and P0 (jω0 ) = R0 + jI0 . Then it is easy to show that 2  2  I0 α(1 + |P0 |2 − α) R0 + I− = , if α = 6 1 R− 1−α 1−α (1 − α)2 where α = r2 (1 + |P0 |2 ). This means that a ball of uncertainty on the ν-gap metric corresponds to a (large) ball of uncertainty on the Nyquist diagram. Figure 17.8 shows 372 GAP METRIC AND ν-GAP METRIC a circle of chordal radius 0.2 centered at the stereographical projection of P0 (jω0 ) = 1 and the corresponding circle on the Nyquist diagram. Figure 17.9 and Figure 17.10 illustrate the uncertainty on the Nyquist diagram corresponding to the balls of uncertainty on the Riemann sphere centered at p0 with chordal radius 0.2. For example, an uncertainty of 0.2 at |p0 (jω0 )| = 1 for some ω0 (i.e., δν (p0 , p) ≤ 0.2) implies that 0.661 ≤ |p(jω0 )| ≤ 1.513 and the phase difference between p0 and p is no more than 23.0739o at ω0 . 1 0.8 0.6 0.4 0.2 0 −1 0 1 1 0.5 0 2 3 −0.5 −1 imaginary axis real axis Figure 17.9: Uncertainty on the Riemann sphere and the corresponding uncertainty on the Nyquist diagram Note that kΨ(P1 , P2 )k∞ on its own without the winding number condition is useless for the study of feedback systems. This is illustrated through the following example. Example 17.8 Consider s−1−ǫ . s−1 It is clear that P2 becomes increasingly difficult to stabilize as ǫ → 0 due to the near unstable pole/zero cancellation. In fact, any stabilizing controller for P1 will destabilize all P2 for ǫ sufficiently small. This is confirmed by noting that bobt (P1 ) = 1, bobt (P2 ) ≈ ǫ/2, and δg (P1 , P2 ) = δν (P1 , P2 ) = 1, ǫ ≥ −2. P1 = 1, However, kΨ(P1 , P2 )k∞ = problem. √ |ǫ| 4+4ǫ+2ǫ2 ≈ P2 = ǫ 2 in itself fails to indicate the difficulty of the 17.4. Extended Loop-Shaping Design 373 1 0.5 Imaginary Axis 23.0739 degree 0 p0=0.1 p0=0.5 p0=1 −0.5 p0=1.5 −1 0 0.5 1 Real Axis 1.5 2 Figure 17.10: Uncertainty on the Nyquist diagram corresponding to the balls of uncertainty on the Riemann sphere centered at p0 with chordal radius 0.2 17.4 Extended Loop-Shaping Design Let P be a family of parametric uncertainty systems and let P0 ∈ P be a nominal design model. We are interested in finding a controller so that we have the largest possible robust stability margin; that is, sup inf bP,K . K P ∈P Note that by Theorem 17.8, for any P1 ∈ P, we have arcsin bP1 ,K (ω) ≥ arcsin bP0 ,K (ω) − arcsin ψ(P0 (jω), P1 (jω)), ∀ω. Now suppose we need inf P ∈P bP,K > α. Then it is sufficient to have arcsin bP0 ,K (ω) − arcsin ψ(P0 (jω), P1 (jω)) > arcsin α, ∀ω, P1 ∈ P; that is, bP0 ,K (ω) > sin (arcsin ψ(P0 (jω), P1 (jω)) + arcsin α) , ∀ω, P1 ∈ P. Let W (s) ∈ H∞ be such that |W (jω)| ≥ sin (arcsin ψ(P0 (jω), P1 (jω)) + arcsin α) , ∀ω, P1 ∈ P. 374 GAP METRIC AND ν-GAP METRIC Then it is sufficient to guarantee |W (jω)| < 1. bP0 ,K (ω) Let P0 = M̃0−1 Ñ0 be a normalized left coprime factorization and note that " # ! I 1 := σ (I + P0 (jω)K(jω))−1 M̃0−1 (jω) . bP0 ,K (ω) K(jω) Then it is sufficient to find a controller so that " # I −1 (I + P0 K) M̃0−1 W K < 1. ∞ The process can be iterated to find the largest possible α. Combining the preceding robust stabilization idea and the H∞ loop-shaping in Chapter 16, one can devise an extended loop-shaping design procedure as follows. (A more advanced loop-shaping procedure can be found in Vinnicombe [1993b].) Design Procedure: Let P be a family of parametric uncertain systems and let P0 be a nominal model. (a) Loop-Shaping: The singular values of the nominal plant are shaped, using a precompensator W1 and/or a postcompensator W2 , to give a desired open-loop shape. The nominal plant P0 and the shaping functions W1 , W2 are combined to form the shaped plant, Ps , where Ps = W2 P0 W1 . We assume that W1 and W2 are such that Ps contains no hidden modes. (b) Compute frequency-by-frequency: f (ω) = sup ψ(Ps (jω), W2 (jω)P (jω)W1 (jω)). P ∈P Set α = 0. (b) Fit a stable and minimum phase rational transfer function W (s) so that |W (jω)| ≥ sin(arcsin f (ω) + arcsin α) ∀ω. (c) Find a K∞ such that β := inf K∞ " I K∞ # (I + P0 K∞ ) −1 M̃0−1 W . ∞ (d) If β ≈ 1, stop and the final controller is K = W1 K∞ W2 . If β ≪ 1, increase α and go back to (b). If β ≫ 1, decrease α and go back to (b). 17.5. Controller Order Reduction 17.5 375 Controller Order Reduction The controller order-reduction procedure described in Chapter 15 can, of course, be applied to the loop-shaping controller design in Chapter 16 and the gap or ν-gap metric optimization here. However, the controller order-reduction in the loop-shaping controller design, or gap metric, or ν-gap metric optimization is especially simple. The following theorem follows immediately from Theorems 17.7 and 17.10. Theorem 17.11 Let P0 be a nominal plant and K0 be a stabilizing controller such that bP0 ,K0 ≤ bobt (P0 ). Let K0 = U V −1 be a normalized coprime factorization and let Û , V̂ ∈ RH∞ be such that " # " # Û U − ≤ ε. V V̂ ∞ Then K := Û V̂ −1 stabilizes P0 if ε < bP0 ,K0 . Furthermore, arcsin bP,K ≥ arcsin bP0 ,K0 − arcsin ε − arcsin β for all {P : δν (P, P0 ) ≤ β}. Hence to reduce the controller order one only needs to approximate the normalized coprime factors of the controller. An algorithm for finding the best approximation is also presented in Vinnicombe [1993a, 1993b]. 17.6 Notes and References This chapter is based on Georgiou and Smith [1990] and Vinnicombe [1993a, 1993b]. Early studies of the gap metric can be found in Zames and El-Sakkary [1980] and ElSakkary [1985]. The pointwise gap metric was introduced by Qiu and Davision [1992b]. 17.7 Problems Problem 17.1 Calculate the gap δg (Pi , Pj ) with P1 = 1 , s+1 P2 = Problem 17.2 Let P = minimizing 1 , s−1 10 τ s+1 , τ P3 = s+2 , (s + 1)2 P4 = ∈ [1, 3] and let P0 = s−2 , (s + 1)2 10 τ0 s+1 . 1 . (s + 1)2 Find the optimal τ0 ∈ [1, 3] min max δg (P, P0 ). τo P5 = τ Problem 17.3 Repeat Problems 17.1–17.2 for the ν-gap metric. 376 GAP METRIC AND ν-GAP METRIC Chapter 18 Miscellaneous Topics This chapter considers two somewhat different problems. The first section gives a brief introduction into the problem of model validation and the second section considers the mixed real and complex µ analysis and synthesis. 18.1 Model Validation A key to the success of the robust control theory developed in this book is to have appropriate descriptions of the uncertain system (whether it is an additive uncertainty model or a general linear fractional model). Then an important question is how one can decide if a model description is appropriate (i.e., how to validate a model). For simplicity of presentation, we have chosen to present the discrete time model validation in this section with the understanding that a continuous time problem can be approximated by fast sampling. Suppose we have modeled a set of uncertain dynamical systems by ∆ := {∆ : ∆ ∈ H∞ , k∆k∞ ≤ 1} where the H∞ norm of a discrete time system ∆ ∈ H∞ is defined as k∆(z)k∞ = sup|z|>1 σ (∆(z)). In order to verify whether this model assumption is correct, some experimental data are collected. For example, let the input to the system be the sequence u = (u0 , u1 , . . . , ul−1 ) and the output y = (y0 , y1 , . . . , yl−1 ). A natural question is whether these data are consistent with our modeling assumption. In other words, does there exist a model ∆ ∈ ∆ such that the output of the ∆ for the period of t = 0, 1, . . . , l − 1 is exactly y = (y0 , y1 , . . . , yl−1 ) with the input u = (u0 , u1 , . . . , ul−1 )? If there does not exist a such ∆, then the model is invalidated. If there exists a such ∆, however, it does not mean that the model is validated but it only means that the model is not validated by this set of data and it may be invalidated by another set of data in the future. Hence it is actually more accurate to say our validation procedure is model invalidation. 377 378 MISCELLANEOUS TOPICS Let ∆ be a stable, causal, linear, time-invariant system with transfer matrix ∆(z) = h0 + h1 z −1 + h2 z −2 + · · · where hi , i = 0, 1, . . . are the matrix Markov parameters. Suppose we have applied the input sequence u = (u0 , u1 , . . . , ul−1 ) to the system and collected the output for the period t = 0, 1, . . . , ℓ − 1, y = (y0 , y1 , . . . , yl−1 ). Then the input and output sequences are related by a Toeplitz matrix:      0 ··· 0 h0 u0 y0       .. u1   y1   . 0  h1 h0    . = .  .   .   .. .. .. .    .   . 0  .  .  . ul−1 yl−1 hl−1 hl−2 · · · h0 This equation shows that, for u0 6= 0 and SISO ∆, the inputs and outputs uniquely determine the first ℓ Markov parameters of the transfer function ∆(z). The model is validated (or more accurately not invalidated) if the remaining Markov parameters can be chosen so that ∆(z) ∈ ∆. The existence of such a choice is the classical tangential Carathéodory-Fejér interpolation problem, for which a solution to the MIMO case can be found in Foias and Frazho [1990, page 195]. We shall state this result in the following theorem. But we shall define some notation first. Let (v0 , v1 , . . . , vℓ−1 , vℓ , vℓ+1 , . . .) be a sequence and let πℓ denote the truncation operator such that πℓ (v0 , v1 , . . . , vℓ−1 , vℓ , vℓ+1 , . . .) = (v0 , v1 , . . . , vℓ−1 ). Let v = (v0 , v1 , . . . , vℓ−1 ) be a sequence of vectors and denote   v0 0 ··· 0   ..  v . 0  v0  1  Tv :=  . . .. ..  .  . 0  .  . vl−1 vl−2 · · · v0 Theorem 18.1 Given u = (u0 , u1 , . . . , ul−1 ) and y = (y0 , y1 , . . . , yl−1 ), there exists a ∆ ∈ H∞ , k∆k∞ ≤ 1 such that y = πℓ ∆u if and only if Ty∗ Ty ≤ Tu∗ Tu . Note that the output of ∆ after time t = ℓ − 1 is irrelevant to the test. The condition Ty∗ Ty ≤ Tu∗ Tu is equivalent to i X j=1 kyj k2 ≤ i X j=1 kuj k2 , i = 0, 1, . . . , ℓ − 1 18.1. Model Validation 379 or kπi yk2 ≤ kπi uk2 , i = 0, 1, . . . , ℓ − 1, which is obviously necessary. In fact, the last condition holds for stable, linear, timek∆uk varying operator ∆ with supu6=0 kuk 2 ≤ 1; see Poolla et al. [1994]. Note that if 2 u0 6= 0, then  Tu is of full column rank and the condition can also be written as 1 σ Ty (Tu∗ Tu )− 2 ≤ 1. Using the above theorem, we can derive solutions to some model validation problems easily. For example, consider a set of additive models shown in Figure 18.1. d ❄ ∆(z) ✛ D(z) y ✛ ❄ f ✛ ❄ f ✛ W (z) ✛ P (z) ✛ u Figure 18.1: Model validation for additive uncertainty In this case, y = (P + ∆W )u + Dd, k∆k∞ ≤ 1 where P (z), W (z), D(z) and ∆(z) are causal, linear, time-invariant systems (but not necessarily stable). The disturbance d is assumed to come from a convex set, d ∈ Dconvex ; for example, Dconvex = {d : d ∈ ℓ2 [0, ∞), kdk2 ≤ 1}. For simplicity, we shall also assume that W (∞) is of full column rank. Let D(z) = D0 + D1 z −1 + D2 z −2 + · · · . Theorem 18.2 Given a set of input-output data uexpt = (u0 , u1 , . . . , uℓ−1 ) with u0 6= 0, yexpt = (y0 , y1 , . . . , yℓ−1 ) for the additive perturbed uncertainty system with an additive disturbance d ∈ Dconvex , where Dconvex is a convex set, let û = (û0 , û1 , . . . , ûℓ−1 ) = πℓ (W uexpt ) ŷ = (ŷ0 , ŷ1 , . . . , ŷℓ−1 ) = yexpt − πℓ P uexpt . Then there exists a ∆ ∈ H∞ , k∆k∞ ≤ 1 such that yexpt = πℓ ((P + ∆W )uexpt + Dd) for some d ∈ Dconvex if and only if there exists a d = (d0 , d1 , . . . , dl−1 ) ∈ πℓ Dconvex such that i h σ (Tŷ − TD Td )(Tû∗ Tû )−1/2 ≤ 1 380 MISCELLANEOUS TOPICS where  D0   D 1  TD :=  .  .  . Dl−1 0 D0 .. . Dl−2 ··· .. . .. . ··· 0   0   .  0  D0 Proof. Note that the system input-output equation can be written as (y − P u) − Dd = ∆(W u). Since P, W, D, and ∆ are causal, linear, and time invariant, we have πℓ Dd = πℓ Dπℓ d, πℓ (y − P u) = yexpt − πℓ P πℓ u = yexpt − πℓ P uexpt and πℓ W u = πℓ W πℓ u = πℓ W uexpt . Denote dˆ = (dˆ0 , dˆ1 , . . . , dˆℓ−1 ) = πℓ (Dd). Then it is easy to show that       and Td̂ = TD Td . Now note that dˆ0 dˆ1 .. . ˆ dℓ−1        = TD      d0 d1 .. . dℓ−1       Tπℓ (y−P u−Dd) = Tπℓ (y−P u) − Tπℓ (Dd) = Tŷ − TD Td , Tπℓ W u = Tû and πℓ ∆W u = πℓ ∆πℓ (W u) since ∆ is causal. Applying Theorem 18.1, there exists a ∆ ∈ H∞ , k∆k∞ ≤ 1 such that πℓ [(y − P u) − Dd] = πℓ ∆(W u) = πℓ ∆πℓ (W u) if and only if (Tŷ − TD Td )∗ (Tŷ − TD Td ) ≤ Tû∗ Tû which is equivalent to h i 1 σ (Tŷ − TD Td )(Tû∗ Tû )− 2 ≤ 1. Note that Tû is of full column rank since W (∞) is of full column rank and u0 6= 0, which implies û0 = 6 0. ✷ Note that inf d∈Dconvex h i 1 σ (Tŷ − TD Td )(Tû∗ Tû )− 2 ≤ 1 is a convex problem and can be checked numerically. 18.2. Mixed µ Analysis and Synthesis 381 Many other classes of model validation problems can be solved analogously. For example, consider a coprime factor model validation problem with y = (M + ∆M WM )−1 (N + ∆N WN )u + d whereh M, N, WM , iand WN are causal, linear, time-invariant systems, and ∆M , ∆N ∈ ≤ 1. Then the problem can be solved by multiplying M +∆M WM H∞ , ∆M ∆N ∞ from the left of the system equation and rewriting the system equation as # " i W (d − y) h M . (M y − N u − M d) = ∆M ∆N WN u The model validation of a general LFT uncertainty system is considered in Davis [1995] and Chen and Wang [1996]. For continuous time model validation, see Rangan and Poolla [1996] and Smith and Dullerud [1996]. 18.2 Mixed µ Analysis and Synthesis In Chapter 10, we considered analysis and synthesis of systems with complex uncertainties. However, in practice, many systems involve parametric uncertainties that are real (for example, the uncertainty about a spring constant in a mechanical system). In this case, one has to cover this real parameter variation with a complex disk in order to use the complex µ analysis and synthesis tools, which usually results in a conservative solution. In this section, we shall consider briefly the analysis and synthesis problems with possibly both real parametric and complex uncertainties. The mixed real and complex µ involves three types of blocks: repeated real scalar, repeated complex scalar, and full blocks. Three nonnegative integers, Sr , Sc , and F , represent the number of repeated real scalar blocks, the number of repeated complex scalar blocks, and the number of full blocks, and they satisfy Sr X i=1 ki + Sc X i=1 ri + F X mj = n. j=1 The ith repeated real scalar block is ki × ki , the jth repeated complex scalar block is rj × rj , and the ℓth full block is mℓ × mℓ . The admissible set of uncertainties ∆ ⊂ Cn×n is defined as   ∆ = diag φ1 Ik1 , . . . , φsr Iksr , δ1 Ir1 , . . . , δsc Irsc , ∆1 , . . . , ∆F ] : φi ∈ R, δj ∈ C, ∆ℓ ∈ Cmℓ ×mℓ . (18.1) n×n The mixed µ is defined in the same way as for the complex µ: Let M ∈ C µ∆ (M ) := (min {σ(∆) : ∆ ∈ ∆, det (I − M ∆) = 0})−1 ; then (18.2) 382 MISCELLANEOUS TOPICS unless no ∆ ∈ ∆ makes I −M ∆ singular, in which case µ∆ (M ) := 0. Or, equivalently, 1 µ∆ (M ) := inf {α : det(I − αM ∆) = 0, σ(∆) ≤ 1, ∆ ∈ ∆} . Let ρR (M ) be the real spectral radius (i.e., the largest magnitude of the real eigenvalues of M ). For√example, if a 4 × 4 matrix M has eigenvalues 1 ± j3, −2, 1, then ρ(M ) = |1 + j3| = 10 and ρR (M ) = | − 2| = 2. It is easy to see that µ∆ (M ) = max ρR (M ∆) ∆∈B∆ where B∆ := {∆ : ∆ ∈ ∆, σ(∆) ≤ 1}. Note that max ρR (M ∆) = max ρ(M ∆) if ∆∈B∆ ∆∈B∆ sr = 0. [This should not be confused with the fact that, for a given matrix ∆ ∈ B∆ π and M , ρR (M ∆) may not be equal to ρ(M ∆). For example, M = 2ej 4 and ∆ = 1; then ρ(M ∆) = 2 but ρR (M ∆) = 0 since M has no real eigenvalues. However, one can π choose another ∆1 = e−j 4 such that ρR (M ∆1 ) = 2 = ρ(M ∆).] Define Q = {∆ ∈ ∆ : φi ∈ [−1, 1], |δi | = 1, ∆i ∆∗i = Imi } h i ( ) diag D̃1 , . . . , D̃sr , D1 , . . . , Dsc , d1 Im1 , . . . , dF −1 ImF −1 , ImF : D = . D̃i ∈ Cki ×ki , D̃i = D̃i∗ > 0, Di ∈ Cri ×ri , Di = Di∗ > 0, dj ∈ R, dj > 0  G = diag [G1 , . . . , Gsr , 0, . . . , 0] : Gi = G∗i ∈ Cki ×ki . It was shown in Young [1993] that µ∆ (M ) = max ρR (QM ). Q∈Q Note that the above maximization is not necessarily achieved on the vertices for the real parameters; hence one must search over the entire interval for each real parameter. Again this maximization problem can have many local maximums and a power algorithm has been developed in Young [1993] to compute a lower bound. It should also be noted that even though the complex µ (i.e., sr = 0) is a continuous function of the data, the mixed µ (i.e., sr 6= 0) may only be upper semicontinuous; see Packard and Pandey [1993]. It was also shown in Braatz et al. [1994] and Toker and Özbay [1995] that the computation of µ is a NP hard problem, which means that it may not be computable in a polynomial time. Of course, it should not be interpreted as every µ problem will not be solvable in a polynomial time; merely some might not. Obviously, the upper bound for the complex µ can be applied for the mixed µ when the intervals of the real parameters are covered by complex disks. However, a better bound can be obtained for the mixed µ by exploiting the phase information of the real parameters. To motivate the improved bound for the mixed µ, we consider again the upper bound for the complex µ problem. It is known that µ∆ (M ) ≤ inf σ(DM D−1 ). D∈D 18.2. Mixed µ Analysis and Synthesis 383 This bound can be reformulated using linear matrix inequalities by noting the following: σ(DM D−1 ) ≤ β ⇐⇒ (DM D−1 )∗ DM D−1 ≤ β 2 I ⇐⇒ M ∗ D∗ DM − β 2 D∗ D ≤ 0. Since D is nonsingular and D∗ D ∈ D, we have  µ∆ (M ) ≤ inf min β : M ∗ DM − β 2 D ≤ 0 . D∈D β The following upper bound for the mixed µ was derived by Fan, Tits, and Doyle [1991] and reformulated in the current form by Young [1993]. Theorem 18.3 Let M ∈ Cn×n and ∆ ∈ ∆. Then  µ∆ (M ) ≤ inf min β : M ∗ DM + j(GM − M ∗ G) − β 2 D ≤ 0 . D∈D,G∈G β Proof. Suppose we have a Q ∈ Q such that QM has a real eigenvalue λ ∈ R. Then there is a vector x ∈ Cn such that QM x = λx. 1 2 1 2 1 Let D ∈ D. Then D ∈ D, D Q = QD 2 and 1 1 1 D 2 QM x = QD 2 M x = λD 2 x. Since σ(Q) ≤ 1, it follows that 1 λ2 D 2 x 2 1 = QD 2 M x 2 1 ≤ D 2 Mx 2 . Hence x∗ (M ∗ DM − λ2 D)x ≥ 0. Next, let G ∈ G and note that Q∗ G = QG = GQ; then  ∗ 1 1 1 x∗ GM x = QM x GM x = x∗ M ∗ Q∗ GM x = x∗ M ∗ QGM x λ λ λ = 1 1 ∗ ∗ x M GQM x = x∗ M ∗ G(QM x) = x∗ M ∗ Gx. λ λ That is, x∗ (GM − M ∗ G)x = 0. Note that j(GM − M ∗ G) is a Hermitian matrix, so it follows that for such x x∗ (M ∗ DM + j(GM − M ∗ G) − λ2 D)x ≥ 0. It is now easy to see that if we have D ∈ D, G ∈ G and 0 ≤ β ∈ R such that M ∗ DM + j(GM − M ∗ G) − β 2 D ≤ 0 384 MISCELLANEOUS TOPICS then |λ| ≤ β, and hence µ∆ (M ) ≤ β. ✷ This upper bound has an interesting interpretation: covering the uncertainties on the real axis using possibly off-axis disks. To illustrate, let M ∈ C be a scalar and ∆ ∈ [−1, 1]. We can cover this real interval using a disk as shown in Figure 18.2. The off-axis disk can be expressed as s  2 G G ˜ ˜ ∈ C, ∆ ˜ ≤ 1. j + 1+ ∆, ∆ β β Im Im j G β j Re 1 -1 1 -1 Re -j Centered Disk Off-Axis Disk Figure 18.2: Covering real parameters with disks Hence 1 − ∆ M β 6= 0 for all ∆ ∈ [−1, 1] is guaranteed if  G 1 − j + β s 1+ r  1+ ⇐⇒ 1 − G β  2 G β ˜  M 6= 0, ∆ β M β M 1−jG β β r  2 G  1+ β ⇐⇒   1−jG M β β ⇐⇒  2 M β ˜ 6= 0, ∆ ˜ ∈ C, ∆ ˜ ≤1 ∆ ˜ ∈ C, ∆ ˜ ≤1 ∆ ∗  r  2 G   1+ β     1−jG M β β M β   ≤1  M∗ M GM M∗ G + j( − )−1≤0 β β β β β β ⇐⇒ M ∗ M + j(GM − M ∗ G) − β 2 ≤ 0. 18.2. Mixed µ Analysis and Synthesis 385 The scaling G allows one to exploit the phase information about the real parameters so that a better upper bound can be obtained. We shall demonstrate this further using a simple example. Example 18.1 Let G(s) = s2 + 2s + 1 . s3 + s2 + 2s + 1 1 0.5 0 −0.5 −1 Nyquist diagram 1/G disk centered at (0,0) −1.5 disk centered at (0,−0.2j) disk centered at (0, −j) −2 −1.5 −1 −0.5 0 0.5 1 1.5 Figure 18.3: Computing the real stability margin by covering with disks We are interested in finding the largest k such that 1 + ∆G(s) has no zero in the right-half plane for all ∆ ∈ [−k, k]. Of course, the largest k can be found very easily by using well-known stability test, which gives kmax =  −1  −1 sup µ∆ (G(jω)) = sup max ρR (φG(jω)) ω φ∈[−1,1] ω  −1  = sup {|G(jω)| : ℑG(jω) = 0} = inf ω ω 1 : ℑG(jω) = 0 |G(jω)|  = 0.5. 386 MISCELLANEOUS TOPICS Now we use the complex covering idea to find the best possible k. Note that we only need to find the smallest |∆| so that 1 + ∆G(jω0 ) = 0 for some ω0 or, equivalently, ∆ + 1/G(jω0 ) = 0. The frequency response of 1/G and the disks covering an interval [−k, k] are shown in Figure 18.3. It is clear that a centered disk would give k = 1/ kGk∞ = 0.2970 and an off-axis disk centered at (0, −0.2j) would give k = 0.3984 while an off-axis disk centered at (0, −j) would give the exactly value k = 0.5. The following alternative characterization of the upper bound is useful in the mixed µ synthesis. Theorem 18.4 Given β > 0, there exist D ∈ D and G ∈ G such that M ∗ DM + j(GM − M ∗ G) − β 2 D ≤ 0 if and only if there are D1 ∈ D and G1 ∈ G such that σ    1 D1 M D1−1 − jG1 (I + G21 )− 2 ≤ 1. β Proof. Let D = D12 and G = βD1 G1 D1 . Then M ∗ DM + j(GM − M ∗ G) − β 2 D ≤ 0 ⇐⇒ M ∗ D12 M + j(βD1 G1 D1 M − βM ∗ D1 G1 D1 ) − β 2 D12 ≤ 0 ⇐⇒ (D1 M D1−1 )∗ (D1 M D1−1 ) + j(βG1 D1 M D1−1 − β(D1 M D1−1 )∗ G1 ) − β 2 I ≤ 0 ⇐⇒  D1 M D1−1 − jG1 β ⇐⇒ σ  ∗  D1 M D1−1 − jG1 β  − (I + G21 ) ≤ 0   D1 M D1−1 2 − 12 ≤ 1. − jG1 (I + G1 ) β ✷ Similarly, the following corollary can be shown. Corollary 18.5 µ∆ (M ) ≤ rβ if there are D1 ∈ D and G1 ∈ G such that σ    D1 M D1−1 2 − 12 ≤ r ≤ 1. − jG1 (I + G1 ) β 18.2. Mixed µ Analysis and Synthesis 387 Proof. This follows by noting that    1 D1 M D1−1 σ − jG1 (I + G21 )− 2 ≤ r ≤ 1 β =⇒ Let G2 =  D1 M D1−1 G1 −j rβ r ∗  G1 D1 M D1−1 −j rβ r  ≤I+ G21 ≤I+  G1 r 2 . G1 ∈ G. Then r  ∗   D1 M D1−1 D1 M D1−1 − jG2 − jG2 ≤ I + G22 rβ rβ =⇒ σ    D1 M D1−1 2 − 12 ≤1 − jG2 (I + G2 ) rβ =⇒ µ∆ (M ) ≤ rβ. ✷ Note that this corollary is not necessarily true if r > 1. It is fairly easy to check that the well-posedness condition, main loop theorem, robust stability, and robust performance theorems for the mixed µ setup are exactly the same as the ones for complex µ problems. We are now in the position to consider the synthesis problem with mixed uncertainties. Consider again the general system diagram in Figure 18.4. By the robust performance condition, we need to find a stabilizing controller K so that min sup µ∆ (Fℓ (P, K)) ≤ β. K z ✛ ω P ✛ ✛ w ✲ K Figure 18.4: Synthesis framework By Theorems 18.3 and 18.4, µ∆ (Fℓ (P (jω), K(jω))) ≤ β, ∀ω if there are frequencydependent scaling matrices Dω ∈ D and Gω ∈ G such that    Dω (Fℓ (P (jω), K(jω))) Dω−1 2 − 21 ≤ 1, ∀ω. − jGω (I + Gω ) sup σ β ω 388 MISCELLANEOUS TOPICS Similar to the complex µ synthesis, we can now describe a mixed µ synthesis procedure that involves D, G − K iterations. D, G − K Iteration: (1) Let K be a stabilizing controller. Find initial estimates of the scaling matrices Dω ∈ D, Gω ∈ G and a scalar β1 > 0 such that    Dω (Fℓ (P (jω), K(jω))) Dω−1 2 − 12 ≤ 1, ∀ω. − jGω (I + Gω ) sup σ β1 ω Obviously, one may start with Dω = I, Gω = 0, and a large β1 > 0. (2) Fit the frequency response matrices Dω and jGω with D(s) and G(s) so that D(jω) ≈ Dω , G(jω) ≈ jGω , ∀ ω. Then for s = jω    1 Dω (Fℓ (P (jω), K(jω))) Dω−1 − jGω (I + G2ω )− 2 sup σ β1 ω    D(s) (Fℓ (P (s), K(s))) D−1 (s) ∼ − 12 . − G(s) (I + G (s)G(s)) ≈ sup σ β1 ω (3) Let D(s) be factorized as D(s) = Dap (s)Dmin (s), ∼ Dap (s)Dap (s) = I, −1 Dmin (s), Dmin (s) ∈ H∞ . That is, Dap is an all-pass and Dmin is a stable and minimum phase transfer matrix. Find a normalized right coprime factorization ∼ Dap (s)G(s)Dap (s) = GN G−1 M , GN , GM ∈ H∞ such that ∼ G∼ M GM + GN GN = I. Then ∼ ∼ −1 ∼ G−1 Dap (G−1 M Dap (I + G G) M ) =I and, for each frequency s = jω, we have    D(s) (Fℓ (P (s), K(s))) D−1 (s) ∼ − 12 − G(s) (I + G (s)G(s)) σ β1    −1 Dmin (Fℓ (P, K)) Dmin ∼ ∼ − 21 ∼ − Dap GDap Dap (I + G G) =σ β1 18.3. Notes and References 389   −1 Dmin (Fℓ (P, K)) Dmin ∼ ∼ − 12 D (I + G G) − GN G−1 ap M β1    −1 Dmin (Fℓ (P, K)) Dmin GM −1 ∼ ∼ − 12 =σ − GN GM Dap (I + G G) β1   −1 Dmin (Fℓ (P, K)) Dmin GM − GN . =σ β1 =σ  (4) Define Pa = " Dmin (s) I # P (s) " −1 Dmin (s)GM (s) I # − β1 " GN 0 # and find a controller Knew minimizing kFℓ (Pa , K)k∞ . (5) Compute a new β1 as β1 = sup inf ω D̃ω ∈D,G̃ω ∈G where Γ := σ " D̃ω Fℓ (P, Knew )D̃ω−1 − j G̃ω β(ω) (6) Find D̂ω and Ĝω such that " inf D̂ω ∈D,Ĝω ∈G σ {β(ω) : Γ ≤ 1} ! (I + D̂ω Fℓ (P, Knew )D̂ω−1 − j Ĝω β1 ! 1 G̃2ω )− 2 (I + # . 1 Ĝ2ω )− 2 # . (7) Compare the new scaling matrices D̂ω and Ĝω with the previous estimates Dω and Gω . Stop if they are close, else replace Dω , Gω and K with D̂ω , Ĝω and Knew , respectively, and go back to step (2). 18.3 Notes and References The model validation problems are discussed in Smith and Doyle [1992] in the frequency domain; in Poolla, Khargonekar, Tikku, Krause, Nagpal [1994 in the discrete time domain (on which Section 18.1 is based); and in Rangan and Poolla [1996] and Smith and Dullerud [1996] in the continuous time domain. See also Davis [1995] and Chen and Wang [1996]. The mixed µ problems are discussed in detail in Young [1993] (on which Section 18.2 is based), Fan, Tits, and Doyle [1991], Packard and Pandey [1993], and references therein. 390 18.4 MISCELLANEOUS TOPICS Problems Problem 18.1 Write a Matlab program for the additive model validation problem and try it on a simple experiment in your laboratory. Bibliography [1] Al-Saggaf, U. M. and G. F. Franklin (1987). “An error bound for a discrete reduced order model of a linear multivariable system,” IEEE Trans. Automat. Contr., Vol. AC-32, pp. 815-819. [2] Al-Saggaf, U. M. and G. F. Franklin (1988). “Model reduction via balanced realizations: an extension and frequency weighting techniques,” IEEE Trans. Automat. Contr., Vol. AC-33, No. 7, pp. 687-692. [3] Anderson, B. D. O. (1967). “An algebraic solution to the spectral factorization problem,” IEEE Trans. Automat. Contr., Vol. AC-12, pp. 410-414. [4] Anderson, B. D. O. (1993). “Bode Prize Lecture (Control design: moving from theory to practice),” IEEE Control Systems, Vol. 13, No.4, pp. 16-25. [5] Anderson, B. D. O. and Y. Liu (1989). “Controller reduction: concepts and approaches, ” IEEE Trans. Automat. Contr. , Vol. AC-34, No. 8, pp. 802-812. [6] Anderson, B. D. O. and J. B. Moore (1989). Optimal Control: Linear Quadratic Methods. Prentice Hall, Englewood Cliffs, New Jersey. [7] Anderson, B. D. O. and S. Vongpanitlerd (1973). Network Analysis and Synthesis: A Modern System Theory Approach. Prentice Hall, Englewood Cliffs, New Jersey. [8] Arnold, W. F. and A. J. Laub (1984). “Generalized Eigenproblem algorithms and software for algebraic Riccati equations,” Proceedings of the IEEE, Vol. 72, No. 12, pp. 1746-1754. [9] Balas, G. (1990). Robust Control of Flexible Structures: Theory and Experiments, PhD. Thesis, California Institute of Technology. [10] Balas, G., J. C. Doyle, K. Glover, A. Packard, and R. Smith (1994). µ-Analysis and Synthesis Toolbox, MUSYN Inc. and The MathWorks, Inc. [11] Başar, T. and P. Bernhard (1991). H∞ -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach. Systems and Control: Foundations and Applications. Birkhäuser, Boston. 391 392 BIBLIOGRAPHY [12] Bernstein, D. S. and W. M. Haddard (1989). “LQG control with an H∞ performance bound: A Riccati equation approach,” IEEE Trans. Automat. Contr., Vol. AC-34, pp. 293-305. [13] Bode, H. W. (1945). Network Analysis and Feedback Amplifier Design, Van Nostrand, Princeton. [14] Boyd, S., V. Balakrishnan, and P. Kabamba (1989). “A bisection method for computing the H∞ norm of a transfer matrix and related problems,” Math. Control, Signals, and Systems, Vol. 2, No. 3, pp. 207-220. [15] Boyd, S. and C. Barratt (1991). Linear Controller Design -Limits of Performance, Prentice Hall, Englewood Cliffs, New Jersey. [16] Boyd, S. and C. A. Desoer (1985). “Subharmonic functions and performance bounds in linear time-invariant feedback systems,” IMA J. Math. Contr. and Info., Vol. 2, pp. 153-170. [17] Boyd, S., L. El Ghaoui, E. Feron, and V. Balakrishnan (1994). Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia. [18] Braatz, R. D., P. M. Young, J. C. Doyle, and M. Morari (1994). “Computational complexity of µ calculation,” IEEE Trans. Automat. Contr., Vol. 39, No. 5, pp. 1000-1002. [19] Brogan, W. L. (1991). Modern Control Theory, 3rd ed., Prentice Hall, Englewood Cliffs, New Jersey. [20] Bruinsma, N. A. and M. Steinbuch (1990). “A fast algorithm to compute the H∞ -norm of a transfer function matrix”, Systems and Control Letters, Vol. 14, pp. 287-293. [21] Bryson Jr., A. E. and Y-C. Ho (1975). Applied Optimal Control, Hemisphere Publishing Corporation. [22] Chen, C. T. (1984). Linear System Theory and Design, Holt, Rinehart and Winston. [23] Chen, J. (1995). “Sensitivity integral relations and design trade-offs in linear multivariable feedback systems,” IEEE Trans. Auto. Contr., Vol. 40, No. 10, pp. 1700-1716. [24] Chen, J. and S. Wang (1996). “Validation of linear fractional uncertain models: solutions via matrix inequalities,” IEEE Trans. Auto. Contr., Vol. 41, No. 6, pp. 844-849. [25] Chen, T. and B. A. Francis (1992). “H2 -optimal sampled-data control,” IEEE Trans. Automat. Contr., Vol. 36, No. 4, pp. 387-397. BIBLIOGRAPHY 393 [26] Chen, X. and K. Zhou (1996). “On the algebraic approach to H∞ control,” 1996 IFAC World Congress, San Francisco, California, pp. 505-510. [27] Chilali, M. and P. Gahinet (1996). “H∞ design with pole placement constraints: an LMI approach,” IEEE Trans. Automat. Contr., Vol. 41, No. 3, pp. 358-367. [28] Christian, L. and J. Freudenberg (1994). “Limits on achievable robustness against coprime factor uncertainty,” Automatica, Vol. 30, No. 11, pp. 1693-1702. [29] Chu, C. C. (1985). H∞ optimization and robust multivariable control. Ph.D. thesis, University of Minnesota. [30] Dahleh, M. and I. J. Diaz-Bobillo (1995). Control of Uncertain Systems: A Linear Programming Approach, Prentice Hall, Englewood Cliffs, New Jersey. [31] Daniel, R. W., B. Kouvaritakis, and H. Latchman (1986). “Principal direction alignment: a geometric framework for the complete solution to the µ problem,” IEE Proceedings, Vol. 133, Part D, No. 2, pp. 45-56. [32] Davis, R. A. (1995). Model Validation for Robust Control, Ph.D. dissertation, University of Cambridge, UK. [33] Desoer, C. A., R. W. Liu, J. Murray, and R. Saeks (1980). “Feedback system design: the fractional representation approach to analysis and synthesis,” IEEE Trans. Automat. Contr., Vol. AC-25, No. 6, pp. 399-412. [34] Desoer, C. A. and M. Vidyasagar (1975). Feedback Systems: Input-Output Properties, Academic Press, New York. [35] Doyle, J. C. (1978). “Guaranteed Margins for LQG Regulators,” IEEE Trans. Automat. Contr., Vol. AC-23, No. 4, pp. 756-757. [36] Doyle, J.C. (1982). “Analysis of feedback systems with structured uncertainties,” IEE Proceedings, Part D, Vol.133, pp. 45-56. [37] Doyle, J. C. (1984). “Lecture notes in advances in multivariable control,” ONR/Honeywell Workshop, Minneapolis. [38] Doyle, J. C. (1985). “Structured uncertainty in control system design”, Proc. IEEE Conf. Dec. Contr., Ft. Lauderdale. [39] Doyle, J. C., B. Francis, and A. Tannenbaum (1992). Feedback Control Theory, Macmillan Publishing Company. [40] Doyle, J. C., K. Glover, P.P . Khargonekar, B. A. Francis (1989). “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr., Vol. AC-34, No. 8, pp. 831-847. Also see 1988 American Control Conference, Atlanta, June 1988. 394 BIBLIOGRAPHY [41] Doyle, J., K. Lenz, and A. Packard (1986). “Design examples using µ synthesis: space shuttle lateral axis FCS during reentry,” IEEE Conf. Dec. Contr., December 1986, pp. 2218-2223. [42] Doyle, J. C., A. Packard and K. Zhou (1991). “Review of LFTs, LMIs and µ,” Proc. IEEE Conf. Dec. Contr., England, pp. 1227-1232 [43] Doyle, J. C., R. S. Smith, and D. F. Enns (1987). “Control of plants with input saturation nonlinearities,” American Control Conference, pp. 1034-1039. [44] Doyle, J. C. and G. Stein (1981). “Multivariable feedback design: Concepts for a classical/modern synthesis,” IEEE Trans. Automat. Contr., Vol. AC-26, pp. 4-16, Feb. 1981. [45] Doyle, J. C., J. Wall and G. Stein (1982). “Performance and robustness analysis for structured uncertainty,” in Proc. IEEE Conf. Dec. Contr., pp. 629-636. [46] Doyle, J. C., K. Zhou, K. Glover, and B. Bodenheimer (1994). “Mixed H2 and H∞ performance objectives II: optimal control,” IEEE Trans. Automat. Contr., Vol. 39, No. 8, pp. 1575-1587. [47] El-Sakkary, A. (1985). “The gap metric: robustness of stabilization of feedback systems,” IEEE Trans. Automat. Contr., Vol. 30, pp. 240-247. [48] Enns, D. (1984a). Model Reduction for Control System Design, Ph.D. dissertation, Department of Aeronautics and Astronautics, Stanford University, Stanford, California. [49] Enns, D. (1984b). “Model reduction with balanced realizations: an error bound and a frequency weighted generalization,” Proc. Conf. Dec. Contr., Las Vegas, Nevada. [50] Fan, M. K. H. and A. L. Tits (1986). “Characterization and efficient computation of the structured singular value,” IEEE Trans. Automat. Contr., Vol. AC-31, No. 8, pp. 734-743. [51] Fan, M. K. H., A. L. Tits, and J. C. Doyle (1991). “Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics,” IEEE Trans. Automat. Contr., Vol. AC-36, No. 1, pp. 25-38. [52] Foias, C. and A. E. Frazho (1990). The commutant Lifting Approach to Interpolation Problems, Birkhäuser. [53] Francis, B. A. (1987). A course in H∞ control theory, Lecture Notes in Control and Information Sciences, Vol. 88, Springer-Verlag. [54] Francis, B. A. and J. C. Doyle (1987). “Linear control theory with an H∞ optimality criterion,” SIAM J. Contr. Optimiz., Vol. 25, pp. 815-844. BIBLIOGRAPHY 395 [55] Franklin, G. F., J. D. Powell, and M. L. Workman (1990), Digital Control of Dynamic Systems, 2nd Ed, Addison-Wesley Publishing company. [56] Freudenberg, J. S. (1986). “The general structured singular problem with two blocks of uncertainty.” Proc. 1986 Allerton Conf., Monticello, Illinois. [57] Freudenberg, J. S. and D. P. Looze (1988). Frequency Domain Properties of Scalar and Multivariable Feedback Systems, Lecture Notes in Contr. and Info. Science, Vol. 104, Springer-Verlag, Berlin. [58] Gahinet, P. (1996). “Explicit controller formulas for LMI-based H∞ synthesis,” Automatica, Vol. 32, No. 7, pp. 1007-1014. [59] Gahinet, P. and P. Apkarian (1994). “A linear matrix inequality approach to H∞ control,” Int. J. Robust and Nonlinear Control, Vol. 4, pp. 421-448. [60] Garnett, J. B. (1981). Bounded Analytic Functions, Academic Press. [61] Georgiou, T. T. (1988). “On the computation of the gap metric,” Systems and Control Letters, Vol. 11, pp. 253-257. [62] Georgiou, T. T. and M. C. Smith (1990). “Optimal robustness in the gap metric,” IEEE Trans. Automat. Contr., Vol. AC-35, No. 6, pp. 673-686. [63] Georgiou, T. T. and M. C. Smith (1992). “Robust stabilization in the gap metric: controller design for distributed plants,” IEEE Trans. Automat. Contr., Vol. AC37, No. 8, pp. 1133-1143. [64] Glover, K. (1984). “All optimal Hankel-norm approximations of linear multivariable systems and their L∞ -error bounds,” Int. J. Contr., Vol. 39, pp. 1115-1193, 1984. [65] Glover, K. (1986a). “Robust stabilization of linear multivariable systems: Relations to approximation,” Int. J. Contr., Vol. 43, No. 3, pp. 741-766. [66] Glover, K. (1986b). “Multiplicative approximation of linear multivariable systems with L∞ error bounds, ” Proc. Amer. Contr. Conf., Seattle, WA, pp. 1705-1709. [67] Glover, K. and J. Doyle (1988). “State-space formulae for all stabilizing controllers that satisfy an H∞ norm bound and relations to risk sensitivity,” Systems and Control Letters, Vol. 11, pp. 167-172. [68] Glover, K. and J. C. Doyle (1989). “A state space approach to H∞ optimal control,” in Three Decades of Mathematical Systems Theory: A Collection of Surveys at the Occasion of the 50th Birthday of Jan C. Willems, H. Nijmeijer and J. M. Schumacher (Eds.), Springer-Verlag, Lecture Notes in Control and Information Sciences, Vol. 135, 1989. 396 BIBLIOGRAPHY [69] Glover, K., D. J. N. Limebeer, J. C. Doyle, E. M. Kasenally, and M. G. Safonov (1991). “A characterization of all solutions to the four block general distance problem,” SIAM J. Contr. Optimiz., Vol. 29, No. 2, pp. 283-324. [70] Glover, K., D. J. N. Limebeer, and Y. S. Hung (1992). “A structured approximation problem with applications to frequency weighted model reduction, ” IEEE Trans. Automat. Contr., Vol. AC-37, No. 4, pp. 447-465. [71] Glover, K. and D. McFarlane (1989). “Robust stabilization of normalized coprime factor plant descriptions with H∞ bounded uncertainty,” IEEE Trans. Automat. Contr., Vol. 34, No. 8, pp. 821-830. [72] Glover, K. and D. Mustafa (1989). “Derivation of the maximum entropy H∞ controller and a state space formula for its entropy,” Int. J. Contr., Vol. 50, pp. 899. [73] Goddard, P. J. and K. Glover (1993). “Controller reduction: weights for stability and performance preservation,” Proceedings of the 32nd Conference on Decision and Control, San Antonio, Texas, December 1993, pp. 2903-2908. [74] Goddard, P. J. and K. Glover (1994). “Performance preserving frequency weighted controller approximation: a coprime factorization approach,” Proceedings of the 33nd Conference on Decision and Control, Orlando, Florida, December 1994. [75] Gohberg, I., and S. Goldberg (1981). Basic Operator Theory. Birkhauser, Boston. [76] Gohberg, I., P. Lancaster, and L. Rodman (1986). “On Hermitian solutions of the symmetric algebraic Riccati equation,” SIAM J. Contr. Optimiz., Vol. 24, No. 6, November 1986, pp. 1323-1334. [77] Golub, G. H., and C. F. Van Loan (1983). Matrix Computations, Johns Hopkind Univ. Press, Baltimore. [78] Green, M. (1988). “A relative error bound for balanced stochastic truncation,” IEEE Trans. Automat. Contr., Vol. AC-33, No. 10, pp. 961-965. [79] Green, M. (1992). “H∞ controller synthesis by J-lossless coprime factorization,” SIAM J. Contr. Optimiz., Vol. 30, No. 3, pp. 522-547. [80] Green, M., K. Glover, D. J. N. Limebeer, and J. C. Doyle (1988), “A J-spectral factorization approach to H∞ control,” SIAM J. Contr. Optimiz., Vol. 28, pp. 1350-1371. [81] Green, M. and D. J. N. Limebeer (1995). Linear Robust Control, Prentice Hall, Englewood Cliffs, New Jersey. [82] Hinrichsen, D., and A. J. Pritchard (1990). “An improved error estimate for reduced-order models of discrete time system,” IEEE Trans. Automat. Contr., AC-35, pp. 317-320. BIBLIOGRAPHY 397 [83] Hoffman, K. (1962). Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, New Jersey. [84] Horn, R. A. and C. R. Johnson (1990). Matrix Analysis, Cambridge University Press, first published in 1985. [85] Horn, R. A. and C. R. Johnson (1991). Topics in Matrix Analysis, Cambridge University Press. [86] Horowitz, I. M. (1963). Synthesis of Feedback Systems, Academic Press, London. [87] Hung, Y. S. (1989).“H∞ -optimal control–Part I: model matching, – Part II: solution for controllers,” Int. J. Contr., Vol. 49, pp. 1291-1359. [88] Hung, Y. S. and K. Glover (1986). “Optimal Hankel-norm approximation of stable systems with first order stable weighting functions,” Systems and Control Letters, Vol. 7, pp. 165-172. [89] Hyde, R. A. and K. Glover (1993). “The application of scheduled H∞ controllers to a VSTOL aircraft,” IEEE Trans. Automat. Contr., Vol. 38, No. 7, pp. 10211039. [90] Jonckheere, E. and L. M. Silverman (1983). “A new set of invariants for linear systems - applications to reduced order compensator design,” IEEE Transactions on Automatic Control, Vol 28, No. 10, pp. 953-964. [91] Kailath, T. (1980). Linear Systems, Prentice Hall, Englewood Cliffs, New Jersey. [92] Kalman, R. E. (1964). “When is a control system optimal?” ASME Trans. Series D: J. Basic Engineering, Vol. 86, pp. 1-10. [93] Kalman, R. E. and R. S. Bucy (1960). “New results in linear filtering and prediction theory,” ASME Trans. Series D: J. Basic Engineering, Vol. 83, pp. 95-108. [94] Khargonekar, P. P., I. R. Petersen, and M. A. Rotea (1988). “H∞ - optimal control with state-feedback,” IEEE Trans. Automat. Contr., Vol. AC-33, pp. 786-788. [95] Khargonekar, P. P., I. R. Petersen, and K. Zhou (1990). “Robust stabilization and H∞ -optimal control,” IEEE Trans. Automat. Contr., Vol. 35, No. 3, pp. 356-361. [96] Khargonekar, P. P. and E. Sontag (1982). “On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings,” IEEE Trans. Automat. Contr., Vol. 27, pp. 627-638. [97] Khargonekar, P. P. and A. Tannenbaum (1985). “Noneuclidean metrics and the robust stabilization of systems with parameter uncertainty,” IEEE Trans. Automat. Contr., Vol. AC-30, pp. 1005-1013. 398 BIBLIOGRAPHY [98] Kimura, H. (1997). Chain-Scattering Approach to H∞ -Control, Birkhäuser, Boston. [99] Kwakernaak, H. and R. Sivan (1972). Linear Optimal Control Systems, WileyInterscience, New York. [100] Kucera, V. (1972). “A contribution to matrix quadratic equations,” IEEE Trans. Automat. Contr., AC-17, No. 3, 344-347. [101] Lancaster, P. and L. Rodman (1995). Algebraic Riccati Equations. Oxford University Press, Oxford. [102] Lancaster, P. and M. Tismenetsky (1985). The Theory of Matrices: with applications, 2nd ed., Academic Press. [103] Latham, G. A. and B. D. O. Anderson (1986). “Frequency weighted optimal Hankel-norm approximation of stable transfer function,” Systems and Control Letters, Vol. 5, pp. 229-236. [104] Laub, S. J. (1980). “Computation of ’balancing’ transformations,” Proc. 1980 JACC, Session FA8-E. [105] Lenz, K. E., P. P. Khargonekar, and John C. Doyle (1987). “Controller order reduction with guaranteed stability and performance, ” American Control Conference, pp. 1697-1698. [106] Limebeer, D. J. N., B. D. O. Anderson, P. P. Khargonekar, and M. Green (1992). “A game theoretic approach to H∞ control for time varying systems,” SIAM J. Contr. Optimiz., Vol. 30, No. 2, pp. 262-283. [107] Liu, K. Z., T. Mita, and R. Kawtani (1990). “Parameterization of state feedback H∞ controllers,” Int. J. Contr., Vol. 51, No. 3, pp. 535-551. [108] Liu, Y. and B. D. O. Anderson (1986). “Controller reduction via stable factorization and balancing,” Int. J. Contr., Vol. 44, pp. 507-531. [109] Liu, Y. and B. D. O. Anderson (1989). “Singular perturbation approximation of balanced systems,” Int. J. Contr., Vol. 50, No. 4, pp. 1379-1405. [110] Liu, Y. and B. D. O. Anderson (1990). “Frequency weighted controller reduction methods and loop transfer recovery,” Automatica, Vol. 26, No. 3, pp. 487-497. [111] Liu, Y., B. D. O. Anderson, and U. Ly (1990). “Coprime factorization controller reduction with Bezout identity induced frequency weighting,” Automatica, Vol. 26, No. 2, pp. 233-249. [112] Lu, W. M., K. Zhou, and J. C. Doyle (1996). “Stabilization of uncertain linear systems and linear multidimensional systems,” IEEE Trans. on Automatic Control, vol. 41, No. 1, pp. 50-65. BIBLIOGRAPHY 399 [113] Luenberger, D. G. (1971). “An introduction to observers,” IEEE Trans. Automat. Contr., Vol. 16, No. 6, pp. 596-602. [114] Maciejowski, J. M. (1989). Multivariable Feedback Design, Addison-Wesley, Reading, Massachusetts. [115] Mageirou, E. F. and Y. C. Ho (1977). “Decentralized stabilization via game theoretic methods,” Automatica, Vol. 13, pp. 393-399. [116] Martensson, K. (1971). “On the matrix Riccati equation,” Information Sciences, Vol. 3, pp. 17-49. [117] McFarlane, D. C., K. Glover, and M. Vidyasagar (1990). “Reduced-order controller design using coprime factor model reduction,” IEEE Trans. Automat. Contr., Vol. 35, No. 3, pp. 369-373. [118] McFarlane, D. C. and K. Glover (1990). Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Vol. 138, Lecture Notes in Control and Information Sciences, Springer-Verlag. [119] McFarlane, D. C. and K. Glover (1992). “A loop shaping design procedure using H∞ synthesis,” IEEE Trans. Automat. Contr., Vol. 37, No. 6, pp. 759-769. [120] Meyer, D. G. (1990). “Fractional balanced reduction: model reduction via fractional representation,” IEEE Trans. Automat. Contr., Vol. 35, No. 12, pp. 13411345. [121] Meyer, D. G. and G. Franklin (1987). “A connection between normalized coprime factorizations and linear quadratic regulator theory,” IEEE Trans. Automat. Contr., Vol. 32, pp. 227-228. [122] Molinari, B. P. (1975). “The stabilizing solution of the discrete algebraic Riccati equation,” IEEE Trans. Automat. Contr., June 1975, pp. 396-399. [123] Moore, B. C. (1981). “Principal component analysis in linear systems: controllability, observability, and model reduction,” IEEE Trans. Automat. Contr., Vol. 26, No. 2, pp. 17-32. [124] Moore, J. B., K. Glover, and A. Telford (1990). “All stabilizing controllers as frequency shaped state estimate feed back,” IEEE Trans. Auto. Contr., Vol. AC35, pp. 203-208. [125] Morari, M. and E. Zafiriou (1989). Robust Process Control. Prentice Hall, Englewood Cliffs, New Jersey. [126] Mullis, C. T. and R. A. Roberts (1976). “Synthesis of minimum roundoff noise fixed point digital filters,” IEEE Trans. Circuits Syst., Vol. 23, pp. 551-562. 400 BIBLIOGRAPHY [127] Mustafa, D. (1989). “Relations between maximum entropy / H∞ control and combined H∞ /LQG control,” Systems and Control Letters, Vol. 12, No. 3, p. 193. [128] Mustafa, D. and K. Glover (1990). Minimum Entropy H∞ Control, Lecture Notes in Control and Information Sciences, Springer-Verlag. [129] Mustafa, D. and K. Glover (1991). “Controller reduction by H∞ -balanced truncation,” IEEE Trans. Automat. Contr., Vol. AC-36, No. 6, pp. 669-682. [130] Naylor, A. W., and G. R. Sell (1982). Linear Operator Theory in Engineering and Science, Springer-Verlag. [131] Nagpal, K. M. and P. P. Khargonekar (1991). “Filtering and smoothing in an H∞ setting,” IEEE Trans. Automat. Contr., Vol. AC-36, No. 2, pp. 152-166. [132] Nehari, Z. (1957). “On bounded bilinear forms,” Annals of Mathematics, Vol. 15, No. 1, pp. 153-162. [133] Nett, C. N., C. A. Jacobson and N. J. Balas (1984). “A connection between statespace and doubly coprime fractional representations,” IEEE Trans. Automat. Contr., Vol. AC-29, pp. 831-832. [134] Nett, C. N. and J. A. Uthgenannt (1988). “An explicit formula and an optimal weight for the 2-block structured singular value interaction measure,” Automatica, Vol. 24, No. 2, pp. 261-265. [135] Packard, A. (1991). Notes on µ∆ . Unpublished lecture notes. [136] Packard, A. (1994). “Gain scheduling via linear fractional transformations.” Systems and Control Letters, Vol. 22, pp. 79-92. [137] Packard, A. and J. C. Doyle (1988a). “Structured singular value with repeated scalar blocks,” Proc. American Control Conference, Atlanta. [138] Packard, A. and J. C. Doyle (1988b). Robust Control of Multivariable and Large Scale Systems, Final Technical Report for Air Force Office of Scientific Research. [139] Packard, A. and J. C. Doyle (1993), “The complex structured singular value,” Automatica, Vol. 29, pp. 71-109. [140] Packard, A. and P. Pandey (1993). “Continuity properties of the real/complex structured singular value,” IEEE Trans. Automat. Contr., Vol. 38, No. 3, pp. 415-428. [141] Packard, A., K. Zhou, P. Pandey, J. Leonhardson, and G. Balas (1992). “Optimal, constant I/O similarity scaling for full information and state feedback control problems,” Systems and Control Letters, Vol. 19, No. 4, pp. 271-280. BIBLIOGRAPHY 401 [142] Paganini, F. (1995). “Necessary and sufficient conditions for robust H2 performance,” Proc. IEEE Conf. dec. Contr., New Orleans, Louisiana, pp. 1970-1975. [143] Paganini, F. (1996). “A set-based approach for white noise modeling,” IEEE Trans. Automat. Contr., Vol. AC-41, No. 10, pp. 1453-1465. [144] Pernebo, L. and L. M. Silverman (1982). ”Model reduction via balanced state space Representation,” IEEE Trans. Automat. Contr., Vol. AC-27, No. 2, pp. 382-387. [145] Petersen, I. R. (1987). “Disturbance attenuation and H∞ optimization: a design method based on the algebraic Riccati equation,” IEEE Trans. Automat. Contr., Vol. AC-32, pp. 427-429. [146] Poolla, K., P. P. Khargonekar, A. Tikku, J. Krause and K. Nagpal (1994). “A time domain approach to model validation,” IEEE Trans. Automat. Contr., Vol. 39, No. 5, pp. 951-959. [147] Poolla, K. and A. Tikku (1995). “Robust performance against time-varying structured perturbations,” IEEE Trans. Automat. Contr., Vol. 40, No. 9, pp. 15891602. [148] Qiu, L. and E. J. Davison (1992a). “Feedback stability under simultaneous gap metric uncertainties in plant and controller,” Systems and Control Letters, Vol. 18, pp. 9-22. [149] Qiu, L. and E. J. Davison (1992b). “Pointwise gap metrics on transfer matrices,” IEEE Trans. Automat. Contr., Vol. 37, No. 6, pp. 741-758. [150] Qiu, L. (1995). “On the robustness of symmetric systems,” Proc. IEEE Conf. Dec. Contr., New Orleans, Louisiana, pp. 2659-2660. [151] Qiu, L., B. Bernhardsson, A. Rantzer, E. J. Davison, P. M. Young, and J. C. Doyle (1995). “A formula for computation of the real stability Radius,” Automatica, Vol. 31, No. 6, pp. 879-890. [152] Ran, A. C. M. and R. Vreugdenhil (1988). “Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems,” Linear Algebra and its Applications, Vol. 99, pp. 63-83. [153] Redheffer, R. M. (1959). “Inequalities for a matrix Riccati equation,” Journal of Mathematics and Mechanics, Vol. 8, No. 3. [154] Redheffer, R. M. (1960). “On a certain linear fractional transformation,” J. Math. and Physics, Vol. 39, pp. 269-286. [155] Rangan, S. and K. Poolla (1996). “Time domain validation for sampled-data uncertainty models,” IEEE Trans. Automat. Contr., Vol. 41, No. 7, pp. 980-991. 402 BIBLIOGRAPHY [156] Rosenbrock, H. H. (1974). Computer-Aided Control System Design. Academic Press, London. [157] Safonov, M. G. (1980). Stability and Robustness of Multivariable Feedback Systems, MIT Press. [158] Safonov, M. G. (1982). “Stability margins of diagonally perturbed multivariable feedback systems,” Proc. IEE, Part D, Vol. 129, No. 6, pp. 251-256. [159] Safonov, M. G. (1984). “Stability of interconnected systems having slope-bounded nonlinearities,”6th Int. Conf. on Analysis and Optimization of Systems, Nice, France. [160] Safonov, M. G. and J. C. Doyle (1984). “Minimizing conservativeness of robustness singular values,” in Multivariable Control, S.G. Tzafestas, Ed., Reidel, New York. [161] Safonov, M. G. and D. J. N. Limebeer, and R. Y. Chiang (1990). “Simplifying the H∞ Theory via Loop Shifting, matrix pencil and descriptor concepts,” Int. J. Contr., Vol 50, pp. 2467-2488. [162] Sampei, M., T. Mita, and M. Nakamichi (1990). “An algebraic approach to H∞ output feedback control problems,” Systems and Control Letters, Vol. 14, pp. 13-24. [163] Scherer, C. (1992). “H∞ control for plants with zeros on the imaginary axis,” SIAM J. Contr. Optimiz., Vol. 30, No. 1, pp. 123-142. [164] Scherer, C. (1992). “H∞ optimization without assumptions on finite or infinite zeros,” SIAM J. Contr. Optimiz., Vol. 30, No. 1, pp. 143-166. [165] Seron, M. M., J. H. Braslavsky, and G. C. Goodwin (1997). Fundamental Limitations in Filtering and Control, Springer, Berlin. [166] Shamma, J. (1994). “Robust stability with time-varying structured uncertainty,” IEEE Trans. Automat. Contr., Vol. 39, No. 4, pp. 714-724. [167] Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control, John Wiley & Sons, New York. [168] Smith, R. S. and J. C. Doyle (1992). “Model validation: a connection between robust control and identification,” IEEE Trans. Automat. Contr., Vol. 37, No. 7, pp. 942-952. [169] Smith, R. S. and G. Dullerud (1996). “Continuous-time control model validation using finite experimental data,” IEEE Trans. Automat. Contr., Vol. 41, No. 8, pp. 1094-1105. [170] Stein, G. and J. C. Doyle (1991). “Beyond singular values and loop shapes,” AIAA Journal of Guidance and Control, Vol. 14, No. 1, pp. 5-16. BIBLIOGRAPHY 403 [171] Stoorvogel, A. A. (1992). The H∞ Control Problem: A State Space Approach, Prentice Hall, Englewood Cliffs, New Jersey. [172] Tadmor, G. (1990). “Worst-case design in the time domain: the maximum principle and the standard H∞ problem,” Mathematics of Control, Systems and Signal Processing, pp. 301-324. [173] Tits, A. L. (1995). “On the small mu theorem,” Proc. American Control Conference, Seattle, Washington, pp. 2870-2872. [174] Tits, A. L. and M. K. H. Fan (1995). “On the small µ theorem,” Automatica, Vol. 31, No. 8, pp. 1199-1201. [175] Toker, O. and H. Özbay (1995). “On the NP-hardness of the purely complex µ computation, analysis/synthesis, and some related problems in multidimensional systems,” Proc. American Control Conference, Seattle, Washington, pp. 447-451. [176] Ushida, S. and H. Kimura (1996). “A counterexample to Mustafa-Glovers’ monotonicity conjecture,” Systems and Control Letters, Vol. 28, No. 3, pp. 129-137. [177] Van Dooren, P. (1981). “A generalized eigenvalue approach for solving Riccati equations,” SIAM J. Sci. Stat. Comput., Vol. 2, No. 2, pp. 121-135. [178] Vidyasagar, M. (1984). “The graph metric for unstable plants and robustness estimates for feedback stability,” IEEE Trans. Automat. Contr., Vol. 29, pp. 403417. [179] Vidyasagar, M. (1985). Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, Massachusetts. [180] Vidyasagar, M. (1988). “Normalized coprime factorizations for non-strictly proper systems,” IEEE Trans. Automat. Contr., Vol. 33, No. 3, pp. 300-301. [181] Vidyasagar, M. and H. Kimura (1986). “Robust controllers for uncertain linear multivariable systems,” Automatica, Vol. 22, pp. 85-94. [182] Vinnicombe, G. (1992a). “On the frequency response interpretation on an indexed L2 -gap metric,” Proceedings of the American Control Conference, Chicago, Illinois, pp. 1133-1137. [183] Vinnicombe, G. (1992b). “Robust Design in the graph topology: a benchmark example,” Proceedings of the American Control Conference, Chicago, Illinois, pp. 2063-2064. [184] Vinnicombe, G. (1993a). “Frequency domain uncertainty and the graph topology,” IEEE Trans. Automat. Contr., Vol. 38, No. 9, pp. 1371-1383. [185] Vinnicombe, G. (1993b). Measuring the Robustness of Feedback Systems, Ph.D. dissertation, Cambridge University. 404 BIBLIOGRAPHY [186] Whittle, P. (1990). Risk-sensitive Optimal Control, John Wiley and Sons, New York. [187] Willems, J.C. (1971). “Least Squares Stationary Optimal Control and the Algebraic Riccati Equation”, IEEE Trans. Automat. Contr., Vol. AC-16, No. 6, pp. 621-634. [188] Willems, J. C. (1981). “Almost invariant subspaces: an approach to high gain feedback design – Part I: almost controlled invariant subspaces,” IEEE Trans. Automat. Contr., Vol. AC-26, pp. 235-252. [189] Willems, J. C., A. Kitapci and L. M. Silverman (1986). “Singular optimal control: a geometric approach,” SIAM J. Contr. Optimiz., Vol. 24, pp. 323-337. [190] Wimmer, H. K. (1985). “Monotonicity of Maximal solutions of algebraic Riccati equations,” Systems and Control Letters, Vol. 5, April 1985, pp. 317-319. [191] Wonham, W. M. (1985). Linear Multivariable Control: A Geometric Approach, third edition, Springer-Verlag, New York. [192] Yang, X. H. and A. Packard (1995). “A low order controller design method,” Proc. IEEE Conf. Dec. Contr., New Orleans, Louisiana, pp. 3068-3073. [193] Youla, D. C. and J. J. Bongiorno, Jr. (1985). “A feedback theory of two-degree-offreedom optimal Wiener-Hopf design,” IEEE Trans. Automat. Contr., Vol. AC-30, No. 7, pp. 652-665. [194] Youla, D. C., H. A. Jabr, and J. J. Bongiorno (1976a). “Modern Wiener-Hopf design of optimal controllers: part I,” IEEE Trans. Automat. Contr., Vol. AC-21, pp. 3-13. [195] Youla, D. C., H. A. Jabr, and J. J. Bongiorno (1976b). “Modern Wiener-Hopf design of optimal controllers: part II,” IEEE Trans. Automat. Contr., Vol. AC21, pp. 319-338. [196] Youla, D. C., H. A. Jabr, and C. N. Lu (1974). “Single-loop feedback stabilization of linear multivariable dynamical plants,” Automatica, Vol. 10, pp. 159-173. [197] Youla, D. C. and M. Saito (1967). “Interpolation with positive-real functions,” Journal of The Franklin Institute, Vol. 284, No. 2, pp. 77-108. [198] Young, N. (1988). An Introduction to Hilbert Space, Cambridge University Press. [199] Young, P. M. (1993). Robustness with Parametric and Dynamic Uncertainty, Ph.D. thesis, California Institute of Technology. [200] Zames, G. (1966). “On the input-output stability of nonlinear time-varying feedback systems, parts I and II,” IEEE Trans. Automat. Contr., Vol. AC-11, pp. 228 and 465. BIBLIOGRAPHY 405 [201] Zames, G. (1981). “Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses,” IEEE Trans. Automat. Contr., Vol. AC-26, pp. 301-320. [202] Zames, G. and A. K. El-Sakkary (1980). “Unstable systems and feedback: The gap metric,” Proc. Allerton Conf., pp. 380-385. [203] Zhang, C. and M. Fu (1996). “A revisit to the gain and phase margins of linear quadratic regulators,” IEEE Trans. Automat. Contr., Vol. 41, No. 10, pp. 15271530. [204] Zhou, K. (1992). “On the parameterization of H∞ controllers,” IEEE Trans. Automat. Contr., Vol. 37, No. 9, pp. 1442-1445. [205] Zhou, K. (1995). “Frequency weighted L∞ norm and optimal Hankel norm model reduction,” IEEE Trans. Automat. Contr., vol. 40, no. 10, pp. 1687-1699, October 1995. [206] Zhou K. and J. Chen (1995). “Performance bounds for coprime factor controller reductions,” Systems and Control Letters, Vol. 26, No. 2, pp. 119-127. [207] Zhou, K., J. C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice Hall, Upper Saddle River, New Jersey. [208] Zhou, K. and P. P. Khargonekar (1988). “An algebraic Riccati equation approach to H∞ optimization,” Systems and Control Letters, Vol. 11, pp. 85-92. [209] Zhou, K., K. Glover, B. Bodenheimer, and J. Doyle (1994). “Mixed H2 and H∞ performance objectives I: robust performance analysis,” IEEE Trans. Automat. Contr., Vol. 39, No. 8, pp. 1564-1574. [210] Zhou, T. and H. Kimura (1992). “Minimal H∞ -norm of transfer functions consistent with prescribed finite input-output data,” in Proc. SICE’ 92, Kumamotp, Japan, pp. 1079-1082. [211] Zhu, S. Q. (1989). “Graph topology and gap topology for unstable systems,” IEEE Trans. Automat. Contr., Vol. 34, No. 8, pp. 848-855. 406 BIBLIOGRAPHY Index additive approximation, 105 additive uncertainty, 131, 142 admissible controller, 222 algebraic Riccati equation, 6, 233 complementarity property, 234 solutions, 15 stability property, 234 stabilizing solution, 234 all-pass dilation, 120 all-pass function, 245 Al-Saggaf, U. M., 126 analytic function, 47 Anderson, B. D. O., 312 Argument Principle, 357 Arnold, W. F., 246 bounded real lemma, 7, 238 Boyd, S., 50, 62, 102, 299 Braatz, R. D., 382 Braslavsky, J. H., 102 Brogan, W., 24, 41 Bruinsma, N. A., 62 Cauchy-Schwarz inequality, 46 Chen, C. T., 41 Chen, J., 102, 344, 381, 389 Chen. T., 54 Chen, X., 299 Chilali, M., 299 Christian, L., 341 co-inner function, 245 complementary inner function, 246 complementary sensitivity, 82 conjugate system, 34 controllability, 3, 27 Gramian, 3, 53, 106 matrix, 29 controllable canonical form, 36 controller parameterization, 221 controller reduction, 8, 305 coprime factor uncertainty, 144, 315 coprime factorization, 4, 71, 228, 315 normalized, 318 Balakrishnan, V., 62, 299 balanced model reduction, 99 additive, 105 error bound, 119 frequency-weighted, 124 multiplicative, 125 relative, 125 stability, 117 balanced realization, 4, 37, 105, 107 Balas, G., 216 Balas, N. J, 77 basis, 11 Bezout identity, 71 bisection algorithm, 57 Bode, H. W., 102 Bode integral, 81 Bode’s gain and phase relation, 94 Bongiorno, J. J., 231 Davis, R. A., 381, 389 Davison, E. J., 341, 354, 375 design limitation, 4, 81 design model, 129 design tradeoff, 81 407 408 Desoer, C. A., 50, 62, 102, 139, 158, 231 detectability, 27, 31 D-G-K iteration, 388 D-K iteration, 214 directed gap, 350 dom(Ric), 234 double coprime factorization, 71 Doyle, J. C., 3, 77, 102, 152, 180, 192, 194, 206, 216, 231, 246, 267, 299, 312, 383, 389 dual system, 34 Dullerud, G., 381, 389 eigenvalue, 12 eigenvector, 12, 234 El Ghaoui, L., 299 El-Sakkary, A., 341, 349, 375 Enns, D., 126, 312 entropy, 286 error bound, 4 Fℓ , 165 Fu , 166 Fan, M. K. H., 201, 217, 383, 389 Feron, E., 299 feedback, 65 filtering, 297 Foias, C., 378 Fourier transform, 49 Francis, B. A., 54, 77, 102, 231, 299 Franklin, G. F., 126 Frazho, A. E., 378 frequency weighting, 85 frequency-weighted balanced reduction, 8, 124 Freudenberg, J. S., 102, 341, 348 Frobenius norm, 18 Gahinet, P., 299 gain, 94 gap metric, 9, 154, 341, 349 generalized eigenvector, 15, 234 INDEX Georgiou, T. T., 334, 341, 350, 375 Gilbert’s realization, 37 Glover, K., 3, 126, 246, 299, 309, 312, 325, 341, 344 Goddard, P. J., 309, 312 Gohberg, I., 62 Goldberg, I., 62 Golub, G. H., 24 Goodwin, G. C., 102 Gramian, 106, 110 graph metric, 341 graph topology, 341 Green, M., 126 Hamiltonian matrix, 56, 233 Hankel singular value, 110 Hardy spaces, 48 Hermitian matrix, 13 H∞ control, 2, 7, 269 loop shaping, 315 singular problem, 294 H∞ filtering, 8, 297 H∞ norm, 2, 55 H∞ optimal controller, 282, 286 H∞ performance, 85, 88 H∞ space, 45, 47, 50 − space, 50 H∞ Hilbert space, 45 Hinrichsen, D., 126 Horn, R. A., 24 Horowitz, I. M., 102, 341 H2 norm, 53 H2 optimal control, 253 H2 performance, 85, 87 H2 space, 45, 47 H2 stability margin, 265 H2⊥ space, 48 Hung, Y. S., 126 Hurwitz, 29 Hyde, R. A., 341 image, 12 INDEX 409 induced norm, 17 inner function, 245 inner product, 46 inner-outer factorization, 248 input sensitivity, 82 integral control, 8, 294 internal model principle, 294 internal stability, 4, 68 invariant subspace, 15 invariant zero, 39, 242 inverse of a transfer function, 35 LQG stability margin, 265 LQG/LTR, 102 LQR porblem, 255 LQR stability margin, 259 L2 norm, 53 L2 space, 48 L2 (−∞, ∞) space, 47 Lu, W. M., 231 Ly, U., 312 Lyapunov equation, 13, 53, 106 Jacobson, V. A., 77 Johnson, C. R., 24 Jonckheere, E., 347 main loop theorem, 197 Martensson, K., 246 matrix Hermitian, 13 inequality, 239 inversion formulas, 13 norm, 16 square root of a, 23 maximum modulus theorem, 47 McFarlane, D. C., 312, 325, 341, 344 minimal realization, 35, 109 minimum entropy controller, 286 Mita, T., 299 mixed µ, 9, 381 modal controllability, 31 modal observability, 31 model invalidation, 377 model reduction, 105 model uncertainty, 65, 129 model validation, 9, 377 Moore, B. C., 126 Moore, J. B., 231 µ, 5, 183 lower bound, 192 synthesis, 213 upper bound, 192 Mullis, C. T., 126 multiplication operator, 50 multiplicative approximation, 125 multiplicative uncertainty, 131, 143 Mustafa, D., 312 Kabamba, P., 62 Kailath, T., 41, 71 kernel, 12 Khargonekar, P. P., 299, 312, 379, 389 Kimura, H., 302 Kitapci, A., 294 Krause, J., 379, 389 Kwakernaak, H., 267 Lancaster, P., 24, 246 Laub, A. J., 246 Lebesgue measure, 46 left coprime factorization, 71 Lenz, K., 312 L∞ norm, 55 L∞ space, 50 Limebeer, D. J. N., 126 linear combination, 11 linear fractional transformation (LFT), 2, 5, 163 linear matrix inequality (LMI), 239, 277 Liu, Y., 312 loop gain, 83 loop shaping, 9, 315, 325 loop transfer matrix, 82 Looze, D. P., 102, 348 410 Nagpal, K. M., 299, 379, 389 Nakamichi, M., 299 Naylor, A. W., 62 Nett, C. N., 77 nominal performance (NP), 137 nominal stability (NS), 137 nonminimum phase zero, 81 norm, 16 normal rank, 38 normalized coprime factorization, 8, 154 loop shaping, 325 ν-gap metric, 9, 154, 349 null space, 12 observability, 3, 27 Gramian, 3, 53, 106 observable canonical form, 36 observable mode, 31 observer, 31 observer-based controller, 31 optimality of H∞ controller, 282 orthogonal complement, 12 orthogonal matrix, 12 output sensitivity, 82 Özbay, H., 382 Packard, A., 180, 192, 216, 299, 314, 382, 389 Pandey, P., 217, 382, 389 Parseval’s relations, 49 PBH (Popov-Belevitch-Hautus) tests, 31 performance limitation, 81 Pernebo, L., 126 phase, 94 plant condition number, 150 Poisson integral, 81, 335, 348 pole, 38 Poolla, K., 379, 381, 389 positive (semi-)definite matrix, 23 positive real, 247 Postlethwaite, I., 102 Pritchard, A. J., 126 INDEX Qiu, L., 220, 341, 354, 375 quadratic performance, 253 Ran, A. C. M., 299 Rangan, S., 381, 389 range, 12 real µ, 381 real spectral radius, 13 realization, 35 balanced, 110 input normal, 113 minimal, 35 output normal, 113 Redheffer star product, 178 reduced-order controller, 8 regulator problem, 253 relative approximation, 125 return difference, 82 RH∞ space, 50 RH− ∞ space, 50 RH2 space, 48 RH⊥ 2 space, 48 Riccati equation, 233 Riccati operator, 234 right coprime factorization, 71 Roberts, R. A., 126 robust performance, 137 H2 performance, 147 H∞ performance, 147, 197 structured, 202 robust stability (RS), 5, 137 structured, 200 robust stabilization, 315 Rodman, L., 246 Safonov, M. G., 132 Saito, M., 215 Sampei, M., 299 Schur complement, 14 Sell, G. R., 62 sensitivity function, 82 Seron, M. M., 102 INDEX Silverman, L. M., 126, 347 singular value decomposition (SVD), 19, 51 singular H∞ problem, 294 singular vector, 20 Sivan, R., 267 skewed performance specification, 150 Skogestad, S., 102 small gain theorem, 3, 129, 137 Smith, M. C., 334, 341, 375 Smith, R. S., 381, 389 span, 11 spectral radius, 12 stability, 27 internal, 68 margin, 265 stabilizability, 27 stabilizable, 29 stabilization, 221 stabilizing controller, 6, 221 stable invariant subspace, 15, 234 star product, 178 Stein, G., 102, 152, 217, 267 Steinbuch, M., 62 strictly positive real, 247 Stoorvogel, A. A., 294, 299 structured singular value, 5, 183 lower bound, 192 upper bound, 192 structured uncertainty, 5, 183 Sylvester equation, 13 Tannenbaum, A., 102 Tikku, A., 379, 389 Tismenetsky, M., 24 Tits, A. L., 201, 217, 383, 389 Toker, O., 382 trace, 12 tradeoff, 81 uncertainty, 1, 65, 129 state space, 171 411 unstructured, 5, 129 unitary matrix, 12 Ushida, S., 302 Van Dooren, P., 246 Van Loan, C. F., 24 Vidyasagar, M., 62, 77, 139, 158, 231, 341, 349 Vinnicombe, G., 312, 341, 349, 366, 375 Vreugdenhil, R., 299 Wall, J., 158 Wang, S., 381, 389 weighted model reduction, 124 weighting function, 4, 85, 89 well-posedness, 66, 167 Willems, J. C., 246 winding number, 357 Wonham, W. M., 41 Yang, X. H., 314 Youla, D. C., 215, 221, 231 Youla parameterization, 224, 228 Young, P. M., 217, 382, 389 Zames, G., 9, 132, 158, 349, 375 zero, 3, 38 Zhou, K., 3, 126, 180, 217, 246, 299, 344 Zhu, S. Q., 341